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BULLETIN  NO.  32 


BUREAU  OF  EDUCATIONAL  RESEARCH 
COLLEGE  OF  EDUCATION 

THE  INTERPRETATION  OF  THE  PROBABLE 

ERROR  AND  THE  COEFFICIENT 

OF   CORRELATION 

By 

Charles  W.  Odell 

Assistant  Director,  Bureau  of  Educational  Research 

( 


THE  UBMM  Of  (Hi 

MAR  14  1927 

HKivERsrrv  w  Illinois 


PRICE  50  CENTS 


PUBLISHED  BY  THE  UNIVERSITY  OF  ILLINOIS.  URBANA 

1926 


370 

lit 


TABLE  OF  CONTENTS 


PAGE 


Preface 5 

i  Chapter  I.    Introduction 7 

Chapter  II.    The  Probable  Error 9 

.Chapter  III.    The  Coefficient  of  Correlation 33 


PREFACE 

Graduate  students  and  other  persons  contemplating  ed- 
ucational research  frequently  ask  concerning  the  need  for 
training  in  statistical  procedures.  They  usually  have  in  mind 
training  in  the  technique  of  making  tabulations  and  calcula- 
tions. This,  as  Doctor  Odell  points  out,  is  only  one  phase, 
and  probably  not  the  most  important  phase,  of  needed  train- 
ing in  statistical  methods.  The  interpretation  of  the  results 
of  calculation  has  not  received  sufficient  attention  by  the 
authors  of  texts  in  this  field.  The  following  discussion  of  two 
derived  measures,  the  probable  error  and  the  coefficient  of 
correlation,  is  offered  as  a  contribution  to  the  technique  of 
educational  research.  It  deals  with  the  problems  of  the 
reader  of  reports  of  research,  as  well  as  those  of  original 
investigators.  The  tabulating  of  objective  data  and  the  mak- 
ing of  calculations  from  the  tabulations  may  be  and  fre- 
quently is  a  tedious  task,  but  it  is  primarily  one  of  routine. 
The  interpreting  of  the  results  of  calculation  is  not  a  routine 
task.  Many  conditions  affect  their  meaning  and  the  research 
worker  constantly  encounters  new  problems  of  interpretation. 
It  is,  however,  possible  to  state  certain  general  principles 
that  will  serve  as  a  guide  in  this  phase  of  educational  research. 

Walter  S.  Monroe,  Director. 
July  7,  1926 


THE  INTERPRETATION  OF  THE  PROBABLE  ERROR 
AND  THE  COEFFICIENT  OF  CORRELATION 

CHAPTER  I 
INTRODUCTION 

Purpose  of  this  bulletin.  One  of  the  most  noticeable  recent  devel- 
opments in  the  field  of  education  has  been  the  extensive  application  of 
statistical  methods  to  the  description  of  educational  conditions  and  the 
solution  of  educational  problems.  Only  a  comparatively  few  years  ago 
conditions  were  portrayed  chiefly  in  terms  of  adjectives  and  other  ex- 
pressions of  quality  or  degree,  but  now  these  have  been  superseded  to  a 
considerable  extent  by  definite  quantitative  terms.  There  are  at  least 
two  reasons  why  everyone  engaged  in  educational  work,  from  the  class- 
room teacher  to  the  research  expert,  should  become  acquainted  with 
certain  commonly  used  formulae,  methods  of  computation,  and  other 
statistical  procedures.  In  the  first  place,  situations  frequently  are  en- 
countered in  which  it  is  desirable  to  make  use  of  statistical  procedures 
for  the  purpose  of  collecting  and  analyzing  data  which  have  a  bearing 
upon  practical  educational  problems.  For  the  great  majority  of  educa- 
tional workers,  however,  it  is  probably  more  important  to  be  able  to 
interpret  correctly  the  numerous  statistical  expressions  and  discussions 
which  are  encountered  in  professional  reading  and  other  work.  It  is 
almost  impossible  to  peruse  a  single  issue  of  an  educational  periodical 
or  a  recent  book  in  the  field  of  education  or  to  attend  an  educational 
meeting  without  seeing  or  hearing  many  statistical  terms  employed. 
Most  of  the  commonly  used  methods  of  computation  can  be  mastered 
by  practically  any  person  of  average  intelligence  and  arithmetical  ability 
within  a  rather  short  time,  but  the  power  of  interpreting  correctly  the 
various  measures  derived  by  statistical  methods  is  not  so  easily  ac- 
quired. The  acquisition  of  this  power  demands  considerable  familiarity 
with  the  concepts  involved  and  this  in  turn  requires  clear  and  critical 
thinking. 

It  is  with  the  second  of  the  two  purposes  mentioned  in  the  preced- 
ing paragraph  chiefly  in  mind  that  the  writer  has  attempted  in  this 
bulletin  to  throw  some  light  on  the  use  and  interpretation  of  two  of  the 
most  frequently  used  statistical  measures,1  the  probable  error2   (com- 


'The  term  "statistical  measure,"  sometimes  shortened  to  "measure,"  is  used  in 
this  bulletin  to  refer  to  a  measure  or  quantitative  expression  which  has  been  derived 
from  a  number  of  data  such  as  scores  or  other  measurements  and  which  summarizes 

[7] 


monly  abbreviated  P.E.),  and  the  coefficient  of  correlation  (commonly 
abbreviated  r),  in  the  hope  that  readers  will  be  helped  in  their  under- 
standing of  the  significance  of  these  terms.  Since  the  methods  of  com- 
puting them  can  be  found  in  many  places,3  their  actual  calculation  will 
not  be  explained  in  detail,  although  the  formulae  for  them  will  be  given. 


or  expresses  in  a  single  numerical  index  some  tendency  of  the  original  data.  All  such 
expressions  as  means,  medians,  modes,  measures  of  deviation,  measures  of  relationship, 
and  so  on  are  statistical  measures.  In  order  to  avoid  confusion  the  term  "measure,"  which 
is  often  used  to  refer  to  the  result  obtained  by  applying  a  measuring  instrument  to  an 
individual  case,  will  not  be  used  in  this  bulletin  in  that  sense,  but  "score"  or  "measure- 
ment" will  be  used  instead. 

'The  term  "probable  error"  (P.E.)  has  come  to  be  generally  used  to  include  both 
the  probable  error  proper  and  the  median  deviation  (abbreviated  Md.D.),  although,  as 
will  be  shown  later  in  the  discussion,  the  latter  is  in  no  real  sense  an  error.  For  this 
reason  and  also  to  avoid  confusion  the  term  probable  error  will  sometimes  be  used 
when  median  deviation  would  be  preferable  from  the  standpoint  of  strict  accuracy  of 
use.  The  reader  should  not  obtain  the  idea,  however,  that  the  writer  believes  it  is 
desirable  to  use  probable  error  instead  of  median  deviation;  he  distinctly  does  not 
believe  so. 

3See: 

Odell,  C.  W.  Educational  Statistics.  New  York:  The  Century  Company,  1925, 
p.  138-39,  150-8-,  221-41,  or  any  other  standard  text  on  statistics. 

[8] 


CHAPTER  II 
THE  PROBABLE  ERROR1 

Formula  for  the  probable  error.  Since  the  probable  error,  as  shown 
by  the  substitute  term  median  deviation,  is  the  median  of  the  deviations 
or  differences  of  the  individual  scores  or  measurements  from  their  aver- 
age,2 it  may  be  computed  simply  by  determining  the  median  of  these 
deviations  or  differences.  However,  the  customary  method  is  to  deter- 
mine the  standard  deviation3  first  and  then  to  multiply  it  by  .67454  to 
obtain  the  probable  error.  In  other  words  the  usual  formula  for  the 
probable  error  is: 

P.E.  =  .6745  a. 
The  relationship  existing  between  the  probable  error  and  the  standard 
deviation  is  therefore  of  the  same  sort  as  that  existing  between  a  foot 
and  a  yard  or  a  pint  and  a  quart,  that  one  always  equals  the  other  mul- 
tiplied by  a  constant  factor.  Thus  just  as  .5  quart  equals  a  pint  and  2 
pints  a  quart,  so  .6745o-  equals  1  P.E.  and  1.48265  P.E.  equals  la. 

Different  uses  of  the  probable  error.  There  are  several  more  or 
less  different  uses  or  meanings  of  the  probable  error,  at  least  five  of 


1The  probable  error,  often  more  properly  called  the  median  deviation,  is  only  one 
of  several  commonly  used  measures  of  the  same  sort.  Among  the  other  similar 
ones  are  the  standard  deviation  (abbreviated  S.D.  or  a  (sigma)  ),  which  in 
certains  uses  becomes  the  standard  error,  the  mean  deviation  (M.D.  or  A.D.), 
the  quartile  deviation  or  semi-interquartile  range  (Q),  and  the  10-90  percentile  range 
(D).  All  of  these  except  the  last  are  rather  frequently  encountered.  In  general,  what- 
ever is  said  about  the  probable  error  may  be  applied  to  these  other  measures  also.  The 
one  important  exception  to  this  statement  is  that  since  these  measures,  except  Q,  differ 
from  the  probable  error  in  magnitude,  their  interpretations  in  numerical  statements 
will,  of  course,  differ.  For  a  discussion  of  these  other  measures  see: 
.  Odell,  op.  cit.,  p.  120-38. 

2The  term  "average"  is  used  here  in  a  general  sense,  that  is,  it  includes  the  arith- 
metic mean,  commonly  called  the  average,  the  median,  the  mode,  and  all  other  measures 
of  central  tendency.  Deviations  or  differences  are  usually  computed  from  the  arithmetic 
mean  but  may  be  taken  from  any  other  measure  of  central  tendency. 

t=t  in  which  .v  denotes  the 

N 

deviation  or  difference  of  a  particular  score  from  the  average,  N  stands  for  the  total 
number  of  cases  or  scores  and  2  (sigma)   is  the  symbol  for  summation. 

4It  is  only  in  the  case  of  a  normal  distribution  or  by  chance  that  the  probable 
error  is  equal  to  nearly  .6745  times  the  standard  deviation.  However,  most  educational 
data  form  distributions  which  approximate  normality  closely  enough  that  no  serious, 
error  is  involved  in  using  the  given  decimal  as  the  multiplier. 

This  number  is  of  course  the  reciprocal  of  .6745. 

[9] 


which  are  fairly  distinct  from  one  another,  and  will  be  dealt  with  in 
this  discussion. 

1.  A  measure  of  the  spread  or  variability  of  a  distribution  of 
data  about  the  average.  When  used  in  this  way  it  should  properly 
be  called  the  median  deviation  (Md.D.).  If  the  term  probable  error 
is  employed  it  should  be  followed  by  the  words  "of  the  distribu- 
tion" and  abbreviated  P.E. 

Dis. 

2.  A  unit  of  measurement.  This  also  is  a  use  for  which  the 
term  median  deviation  is  really  the  correct  one  to  employ,  since  it 
involves  merely  a  particular  use  of  the  median  deviation  of  a  dis- 
tribution. Since  no  subscript  has  been  agreed  upon  to  denote  this 
use,  the  writer  suggests  "U"  for  "unit."  Thus,  when  designating 
the  median  deviation  used  as  a  unit  of  measurement  one  should 
write  Md.D.  .     or  if  one  follows  the  general  practice  rather  than 

the  best,  P.E.    . 

3.  A  measure  of  the  reliability  of  sampling.  The  accepted  ab- 
breviation for  this  use  is  P.E.  with  a  subscript  denoting  the  measure 
to  which  it  applies.  Thus  P.E.  denotes  the  probable  error  of  the 
mean,  P.E.        that  of  the  median,  P.E.    that  of  the  coefficient  of 

Hid .  r 

correlation  and  so  on. 

4.  A  measure  of  the  reliability  or  accuracy  of  any  one  of  a 
number  of  scores  or  measurements  of  the  same  thing.  As  will  be 
explained  later  this  is  from  one  standpoint  a  variety  of  the  imme- 
diately preceding  use.  It  has  no  conventional  abbreviation,  hence 
PE.      is  suggested  as  a  suitable  one. 

5.  A  measure  of  the  reliability  of  a  measuring  instrument.  This 
may  be  divided  into  two  sub-heads  as  follows: 

A.  A  measure  of  the  reliability  or  accuracy  of  scores  ob- 
tained from  a  measuring  instrument  when  compared  with  those 
obtained  from  another  application  of  the  same  or  of  a  suppos- 
edly equivalent  measuring  instrument  to  the  same  individuals. 
This  is  called  the  probable  error  of  estimate  and  is  abbre- 
viated P.E.      . 

Est. 

B.  A  measure  of  the  reliability  or  accuracy  of  scores  ob- 
tained from  a  measuring  instrument  when  compared  with  the 
theoretically  true  scores.  This  is  called  the  probable  error  of 
measurement  and  is  best  abbreviated  P.E.,, 

Meat. 

The  probable  error  as  a  measure  of  the  spread  or  variability  of  a 
distribution  of  data  around  its  average.   As  was  stated  above  the  term 

[10] 


probable  error  is  a  misnomer  in  connection  with  this  use  and  median 
deviation  should  be  used  instead.  Therefore,  the  writer  will  use  the 
latter  expression  in  the  discussion  immediately  following.  The  use  of 
the  median  deviation  as  a  measure  of  the  spread  or  variability  of  a  distri- 
bution of  data  around  its  average  is  the  fundamental  one  and  from  it 
all  the  others  are  derived.  When  a  number  of  scores  or  measurements 
yielded  by  a  test  or  other  measuring  instrument  are  tabulated  in  a  dis- 
tribution it  is  frequently  desirable  and  useful  to  describe  in  some  concise 
way  their  spread  or  variability  about  the  average.  In  other  words,  one 
often  desires  to  indicate  or  summarize  by  a  single  numerical  expression 
the  extent  to  which  the  individual  scores  tend  to  cluster  about  or  depart 
from  their  average.  For  example,  if  the  marks  assigned  the  pupils  in  two 
classes  have  been  tabulated  and  the  averages  of  both  classes  are  com- 
puted and  found  to  be  85  percent,  one  knows  that  the  average  rating 
of  the  classes  is  the  same  but  he  does  not  know  whether  or  not  the 
classes  are  equally  homogeneous  in  regard  to  the  ratings  given.  In  other 
words,  he  does  not  know  whether  all  the  pupils  in  both  classes  received 
marks  closely  grouped  around  the  average,  whether  their  marks  ranged 
from  decidedly  below  to  considerably  above  the  average,  or  whether 
the  first  condition  held  in  one  class  and  the  second  in  the  other. 

One  of  the  measures  most  commonly  used  as  an  index  of  the  amount 
of  spread  or  variability  is  the  median  deviation.6  This  is  exactly  what 
its  name  implies,  the  median  of  the  deviations  or  differences  of  the  indi- 
vidual scores  from  their  average.  Since  the  median  is  a  point  on  each 
side  of  which  there  are  half  of  the  measures  in  the  whole  distribution, 
the  median  deviation  is  always  of  such  a  magnitude  that  half  of  the 
individual  scores  differ  from  their  average  by  less  than  this  amount  and 
half  by  more.  For  example,  if  one  of  the  classes  referred  to  above  had 
a  median  deviation  of  3  percent  it  would  mean  that  half  of  the  pupils' 
marks  were  within  3  percent  of  85,  that  is,  from  82  to  88,  and  the  other 
half  either  below  82  or  above  88.  Similarly  a  median  deviation  of  5 
percent  for  the  other  class  would  mean  that  the  marks  of  half  of  its 
members  were  between  80  and  90  and  those  of  the  other  half  either 
below  80  or  above  90.  From  these  values  of  the  median  deviation,  3  and 
5,  one  would  know  that  the  first  class  was  more  homogeneous  than  the 
second  in  respect  to  the  ratings  given. 


"It  cannot  be  said  in  any  real  sense  that  the  differences  between  the  individual 
scores  or  measures  of  a  number  of  individuals  and  their  average  are  errors.  Despite 
this  fact,  however,  the  term  probable  error  is  frequently  used  in  this  connection. 

[11] 


TABLE  1.8  A  SUMMARY  OF  TABLE  I  OF  JOHNSON'S  STUDY  GIVING  THE 
MEAN   ACCURACY  SCORES   EARNF.D  ON  THE   COURTIS  SUB- 
TRACTION CARD  NO.  33  BY  THE  GROUPS  USING  THE 
SEVERAL  METHODS  OF  SUBTRACTION 


Method 

Score 

I 

II 

III 

IV 

Mixed 

17 

75 

13 

2 

8 

3 

16 

74 

5 

4 

3 

6 

15 

35 

2 

1 

la 

1 

14 

22 

3 

1 

2 

13 

5 

1 

12 

6 

11 

1 

1 

10 

1 

9 

1 

N 

220 

23 

'  8 

13 

13 

M 

15.7  , 

16.2 

15.8 

16.4 

15.5 

Md.D. 

0,9 

0.7 

0.8 

0.6 

1.1 

"Printed  as  5  but  here  taken  as  15,  since  the  use  of  the  latter  value  checks  with 
the  mean  reported. 


The  actual  use  of  the  median  deviation  in  this  way  is  shown  by  the 
following  table  taken  from  a  magazine  article.7  This  table  shows  the 
distributions  of  scores  on  the  Courtis  Subtration  Card  No.  33  made  by 
five  groups  of  pupils  who  had  used  different  methods  of  subtraction. 
Below  each  column  in  the  table  are  given  the  number  of  pupils,  the 
mean  score,  the  standard  deviation  and  the  median  deviation  of  the 


7Rucn.  G.  M.,  Kxight,  F.  B.,  and  Lutes,  O.  S.  "On  the  relative  merits  of  sub- 
traction methods:  another  view,"'  Journal  of  Educational  Research,  11:154-55,  Feb- 
ruary, 1925.  For  other  examples  of  the  use  of  the  probable  error  or  median  deviation 
see  the  following  references: 

Courtis,  S.  A.  The  Gary  Public  Schools:  Measurement  of  Classroom  Products. 
New  York:    General  Education  Board,  1919,  p.  213. 

Stoddard,  G.  D.  '"Iowa  Placement  Examinations."  University  of  Iowa  Studies 
in  Education,  Vol.  3.  No.  2.    Iowa  City:  University  of  Iowa,  1925,  p.  62-64. 

Kallom,  A.  W.  "Times  of  writing  each  of  the  Arabic  numerals  determined  by 
the  reaction  time  method,"  Journal  of  Educational  Psychology,  7:226-28,  April,  1916. 

Childs,  H.  G.  "Measurement  of  the  drawing  ability  of  two  thousand  one  hun- 
dred and  seventy-seven  children  in  Indiana  city  school  systems  by  a  supplemented 
Thorndike  Scale,"  Journal  of  Educational  Psychology,  6:391-408,  September,   1915. 

Tor  purposes  of  convenience  the  tables  in  this  bulletin  are  numbered  consecu- 
tively instead  of  as  in  the  sources  from  which  they  are  quoted.  Also  some  of  them  have 
been  modified  slightly  in  order  to  be  consistent  or  to  follow  the  best  form,  parts  of 
some  have  been  omitted,  and  occasional  errors  have  been  corrected. 

[12] 


distribution  in  that  column.  For  example,  220  pupils  used  the  first 
method,  their  mean  score  was  15.7,  and  the  median  deviation  of  their 
scores  .9.  This  statement  is  merely  a  way  of  expressing  the  fact  that 
half  of  the  scores  probably  fell  within  .9  of  the  mean,  or  between  14.8 
and  16.6,  and  half  outside  of  these  limits.  Similarly,  for  the  pupils  who 
used  the  second  method  the  mean  was  16.2  and  the  median  deviation  .7, 
which  indicates  that  half  of  the  pupils  probably  made  scores  between 
15.5  and  16.9  and  half  lower  or  higher  than  these  limits. 

It  will  perhaps  be  helpful  to  illustrate  the  significance  of  the  median 
deviation  by  a  graphical  representation.  With  this  in  mind  Figure  1  has 
been  prepared.   The  portion  of  the  figure  at  the  left  represents  graph- 


jro.of 
Cases 


b  ■ 
If.. 

2- 
oJ 


^ 


lg1   \K   TS 

Score 


Wo.  of 

Cases 


6  - 
*  ■ 


2" 


~.u 


£J 


IE      Vr      16 

Score 


Data  from  Column  MIVtt      Data  from  Column  "Mixed" 

Figure  1.    Graphical  Representation  of  the  Data  in  the  Last 
Two  Columns  of  Table  I 

ically  the  distribution  of  scores  contained  in  Column  IV  of  Table  I,  the 
portion  at  the  right  the  scores  in  the  column  headed  "Mixed."  The  dis- 
tributions in  these  two  columns  were  chosen  for  graphical  representa- 
tion because  the  total  number  of  scores  in  each  is  the  same  and  there- 
fore the  areas  of  the  surfaces  representing  them  are  equal.  Inspection 
of  the  figure  makes  it  evident  that  the  scores  represented  at  the  right 
spread  out  considerably  more  than  do  the  others.  The  height  of  the 
graph  at  the  right  is  less  and  the  length  of  its  base  greater  than  of  the  one 
at  the  left,  which  indicates  a  wider  spread  of  scores.  This  agrees  with 
the  fact  that  the  median  deviation  of  the  distribution  represented  by 
it  is  1.1,  whereas  that  of  the  other  one  is  only  .6.  It  might  be  noted  also 
that  neither  of  the  graphs  approach  normality  very  closely,  the  one  at 
the  right,  however,  doing  so  more  nearly  than  the  one  at  the  left. 


[13] 


The  interpretation  of  the  median  deviation,  when  used  to  measure 
how  closely  individual  scores  or  measurements  cluster  about  their  aver- 
age or  how  far  they  spread  out  from  it,  may  be  extended  further  than 
has  been  suggested  in  the  preceding  paragraphs  by  stating  what  fraction 
of  the  scores  will  not  differ  from  the  average  by  more  than  a  given  multi- 
ple of  the  median  deviation.  For  the  few  smallest  integral  multiples  we 
may  state  as  follows:9 

50.00  percent  of  scores  differ  from  the  average  by  less  than  1  Md.D. 
82.26  percent  of  scores  differ  from  the  average  by  less  than  2  Md.D. 
95.70  percent  of  scores  differ  from  the  average  by  less  than  3  Md.D. 
99.30  percent  of  scores  differ  from  the  average  by  less  than  4  Md.D. 
99.92  percent  of  scores  differ  from  the  average  by  less  than  5  Md.D. 

We  may  also  change  the  form  of  statement  and  say  that  the  chances  are: 

1      to  1  that  a  score  differs  from  the  average  by  less  than  1  Md.D. 

4.6  to  1  that  a  score  differs  from  the  average  by  less  than  2  Md.D. 

22     to  1  that  a  score  differs  from  the  average  by  less  than  3  Md.D. 

142      to  1  that  a  score  differs  from  the  average  by  less  than  4  Md.D. 

1,340     to  1  that  a  score  differs  from  the  average  by  less  than  5  Md.D}" 


^Although  the  numerical  interpretations  given  in  the  text  hold  exactly  only  in  the 
case  of  normal  frequency  distributions  they  may  be  used  without  serious  error  in  deal- 
ing with  the  large  majority  of  tabulations  of  such  educational  facts  as  pupils'  heights, 
weights,  school  marks  and  test  scores,  teachers'  salaries,  numbers  of  pupils  to  the  room, 
and  so  forth.  For  example,  109  of  the  scores  in  the  first  column  of  Table  I  fall  within 
1  Md.D.  of  the  mean,  whereas  110  would  be  expected  to  do  so. 

"'It  has  been  previously  stated  that  the  chief  difference  between  the  interpretation 
of  the  standard  deviation  (c),  the  mean  deviation  (M.D.),  and  the  10-90  percentile 
range  (D.),  and  of  the  median  deviation  has  to  do  with  numerical  interpretation.  For 
example,  it  is  to  be  expected  that: 

68.27  percent  of  scores  differ  from  the  average  by  less  than   1   a 

95.44  percent  of  scores  differ  from  the  average  by  less  than  2   a 

99.74  percent  of  scores  differ  from  the  average  by  less  than  3   a 

99.99  percent  of  scores  differ  from  the  average  by  less  than  4  a 
Using  the  other  form  of  statement,  the  chances  are: 

2.15  to  1  that  a  score  differs  from  the  average  by  less  than  1  a 

21        to  1  that  a  score  differs  from  the  average  by  less  than  2  a 

369        to  1  that  a  score  differs  from  the  average  by  less  than  3  a 

15,772        to  1  that  a  score  differs  from  the  average  by  less  than  4  a 
Also  it  is  probable  that: 

57.51  percent  of  scores  differ  from  the  average  by  less  than  1  M.D. 

88.94  percent  of  scores  differ  from  the  average  by  less  than  2  M.D. 

98.33  percent  of  scores  differ  from  the  average  by  less  than  3  M.D. 

99.86  percent  of  scores  differ  from  the  average  by  less  than  4  M  .D. 
Or  the  chances  are: 

1.55  to  1  that  a  score  differs  from  the  average  by  less  than   1  M.D. 

8        to  1  that  a  score  differs  from  the  average  by  less  than  2  M.D. 


[14] 


These  more  extended  interpretations  may  be  illustrated  by  re- 
ferring back  to  the  examples  used  earlier.  For  the  first  of  the  two 
classes  referred  to,  which  had  a  mean  score  of  85  and  a  median  devia- 
tion of  3,  it  is  not  only  probable  that  half  of  its  members  have  scores 
between  82  and  88  but  also  that  about  82  percent  of  them  have  scores 
between  79  and  91  (85  ±  6),  almost  96  percent  between  76  and  94 
(85  ±  9),  over  99  percent  between  73  and  97  (85  ±  12),  and  very 
nearly  100  percent  between  70  and  100  (85  ±  15).  Using  the  other 
form  of  statement  for  the  first  column  of  Table  I,  the  chances  are  1  to  1, 
or  even,  that  a  particular  score  chosen  at  random  falls  between  14.8  and 
16.6  (15.7  ±  .9),  4.6  to  1  that  it  falls  between  13.9  and  17.5  (15.7  ±  1.8), 
22  to  1  that  it  is  between  13.0  and  18.4  (15.7  ±  2.7),  142  to  1  that  it  is 
between  12.1  and  19.3  (15.7  ±  3.6),  and  1340  to  1  that  it  is  between 
11.2  and  20.2  (15.7  ±  4.5).11 

The  probable  error  as  a  unit  of  measurement.12  In  dealing  with 
data  of  various  sorts  one  encounters  many  different  units.  The  unit 
usually  used  for  school  marks  is  the  percent,  for  ages  the  year,  the 
month,  or  the  day,  for  salaries  the  dollar,  for  heightsfthe  foot  or  the 
inch,  for  weights  the  pound,  for  spelling  the  word,  iSt  arithmetic  the 
example,  and  so  on.  In  the  case  of  such  characteristics  as  height,  weight, 
age,  salary,  and  so  forth,  even  though  there  are  commonly  used  units 
of  measurement,  it  is  difficult  if  not  impossible  to  compare  one  trait 
with  another.  For  example,  one  cannot  readily  determine  whether  a 
pupil's  height  of  four  feet,  eleven  inches,  his  weight  of  102  pounds,  or 
his  age  of  12  vears  and  8  months  is  the  highest  or  lowest  ranking  when 


59        to  1  that  a  score  differs  from  the  average  by  less  than  3  M.D. 

706        to  1  that  a  score  differs  from  the  average  by  less  than  4  M.D. 
For  the  10-90  percentile  range  the  corresponding  statements  are: 

99  percent  of  scores  differ  from  their  average  by  less  than   1  D. 

99.99997  percent  of  scores  differ  from  their  average  by  less  than  2  D. 
And  the  chances  are: 

95  to  1  that  a  score  differs  from  the  average  by  less  than   1  D. 

3,380,614  to  1  that  a  score  differs  from  the  average  by  less  than  2  D. 

As  was  suggested  previously  the  quartiie  deviation  may  be  interpreted  in  the 
same  way  numerically  as  the  median  deviation. 

"The  fact  that  one,  or  sometimes  even  both,  of  the  limits  within  which  a  certain 
fraction  of  the  scores  may  be  expected  to  fall  comes  outside  the  range  of  actually  ob- 
tained scores  is  due  to  the  fact  that  the  scores  do  not  form  a  normal  distribution.  That 
they  do  not  is  often  caused  by  the  number  of  scores  being  small,  as  well  as  by  causes 
inherent  in  the  nature  of  the  data  themselves. 

13This  use  is  derived  directly  from  the  one  discussed  in  the  preceding  paragraph 
and  also  is  one  to  which  the  name  median  deviation  should  properly  be  applied.  The 
writer  will  therefore  employ  the  latter  term,  abbreviated  Md.D.  ,  throughout  his  treat- 
ment of  this  use.  *■  • 

[15] 


compared  with  other  similar  pupils.  There  are  also  many  situations  in 
which  there  is  no  commonly  used  unit  or  indeed  any  conventional  unit 
closely  connected  with  the  type  of  thing  being  measured.  Probably  most 
of  such  cases  in  the  field  of  education  have  to  do  with  the  measurement 
of  difficulty,  such  as  difficulty  of  examples  in  arithmetic,  of  questions 
in  history  or  geography,  of  passages  in  reading,  of  words  in  spelling, 
and  so  forth. 

To  meet  the  need  for  a  common  unit  in  which  all  scores  and 
measurements,  including  those  for  which  no  conventional  units  are  avail- 
able, may  be  expressed  and  thereby  easily  compared  the  median  devia- 
tion has  been  adopted  and  come  into  rather  common  use.  Irrespective 
of  the  units  in  terms  of  which  scores  or  measurements  have  been  ex- 
pressed originally,  by  applying  certain  statistical  procedures  they  may 
be  expressed  in  terms  of  median  deviations. 

The  most  frequent  use  of  the  median  deviation  as  a  unit  has  proba- 
bly been  in  connection  with  the  construction  of  standardized  educational 
measuring  instruments.  The  values  or  difficulties  assigned  the  different 
items  or  steps  on  the  scale  or  the  distances  between  the  steps  are  very 
frequently  expressed  in  such  units.  An  example  of  this  may  be  found 
in  connection  with  Woody's  Arithmetic  Scales,13  given  as  part  of  his 
account  of  the  derivation  of  these  scales.  Woody  describes  how  the  dif- 
ficulty values  of  the  exercises  composing  each  scale  were  determined. 
The  essential  steps  in  this  determination  consisted  of  finding  the  median 
deviation  of  the  distribution  of  scores14  for  each  scale  and  then  measur- 
ing the  distance  of  each  exercise  from  the  average  of  the  distribution  in 
terms  of  Md.D.  units.13  Different  results  were  obtained  in  the  different 
school  grades  so  that  it  was  necessary  to  combine  these  into  average 
results.   Finally,  Woody  located  zero16  points,  that  is,  points  of  absolute 


"Woody,  Clifford.  "Measurements  of  Some  Achievements  in  Arithmetic."  Teach- 
ers College  Contributions  to  Education,  No.  80.  New  York:  Teachers  College,  Colum- 
bia University,  1916,  p.  29-54. 

"It  is  assumed  that  the  distribution  of  pupils'  scores  represents  the  distribution 
of  their  abilities. 

"It  does  not  seem  necessary  for  the  purpose  of  the  present  discussion  to  explain 
in  complete  detail  just  how  this  was  done.  Briefly,  Woody  found  the  percent  of  pupils 
obtaining  the  correct  answer  to  each  exercise  and,  on  the  assumption  that  the  distribu- 
tion of  pupils'  abilities  was  normal,  calculated  the  degree  of  difficulty  of  an  exercise  in 
terms  of  the  number  of  Md.D.  units  that  the  ability  required  to  do  each  exercise  dif- 
fered from  the  average  ability  of  the  group.  For  a  fuller  explanation  of  the  method  of 
procedure,  see: 

Odell,  op.  cit.,  p.  313-15. 

10The  determination  of  such  zero  points  is  not  a  necessary  part  of  the  process  of 
employing  Md.D.  t    but  merely  renders  the  values  so  expressed  more  usable.   The  actual 

[16] 


lack,  of  ability  to  solve  exercises  in  the  four  fundamental  operations  and 
transformed  the  Md.D.  ,  values  of  the  various  exercises  from  distances 
from  the  averages  of  the  distributions  into  distances  from  the  zero 
points.  To  illustrate  this  simply,  we  may  express  John's  height  by 
saying  that  he  is  six  inches  taller  than  Paul.  If,  however,  we  know  that 
Paul  is  five  feet  and  three  inches  tall  we  can  express  John's  height  much 
more  satisfactorily  for  most  purposes  by  saying  that  he  is  five  feet  and 
nine  inches  above  the  zero  point  which  is,  of  course,  zero  inches  or  no 
height  at  all. 

To  show  the  final  result  of  the  process,  that  is,  the  difficulty  values 
determined  for  the  exercises,  a  portion  of  one  of  Woody's  tables17  is  given 
as  Table  II.   Exercise  2  was  found  to  be  the  easiest,  having  a  difficulty 

value  of  1.23  Md.D.    ,  exercise  3  was  next  with  a  value  of  1.40  Md.D. 

u/  _  u. 

and  so  on  up  to  exercise  38,  the  most  difficult,  which  had  a  value  of 
9.19  Md.D.  ,  .  After  the  difficulty  values  have  been  so  expressed  we  can 
not  only  say,  for  example,  that  exercise  3  is  .17  Md.D  and  exercise  5 
\. 21  Md.D.  T  more  difficult  than  No.  2,  but  also,  if  the  zero  point  has 
been  located  accurately,  that  exercise  5  is  about  twice  as  hard  as  No.  2, 
but  only  half  as  difficult  as  No.  15. 

The  preceding  discussion  has  used  the  term  Md.D.  .    but  perhaps 

not  made  clear  just  what  it  really  means.  Since  50  percent  of  the  scores 
in  a  normal  distribution  fall  within  1  Md.D.  of  the  average  and  since  a 
normal  distribution  is  symmetrical  it  follows  that  half  of  these  50  per- 
cent, or  25  percent,  of  the  scores  will  fall  within  1  Md.D.  of  the  average 
on  each  side.  That  is,  25  percent  will  fall  between  the  average  and  1  Md.D. 


determination  of  zero  points  usually  rests,  at  least  in  part,  upon  opinion  as  to  just  what 
constitutes  absolute  lack  of  ability  in  a  given  field.  Sometimes  it  is  possible  to  deter- 
mine rather  accurately  just  what  is  the  least  difficult  task  of  a  certain  sort  and  to 
locate  that  degree  of  ability  just  barely  insufficient  to  accomplish  this  task,  but  in 
many  cases  this  can  not  or  at  least  has  not  been  done. 

"Woody,  op.  cit.,  p.  54.   Other  examples  of  the  use  of  Md.D.      as  a  unit  may  be 

U. 
found  in: 

Buckingham,  B.  R.  "Spelling  Ability — Its  Measurement  and  Distribution." 
Teachers  College  Contributions  to  Education,  No.  59.  New  York:  Teachers  College, 
Columbia  University,  1913,  p.  40-65. 

Monroe,  W.  S.  An  Introduction  to  the  Theory  of  Educational  Measurements. 
Boston:  Houghton  Mifflin,  1923,  p.  61-62,  94-103,  138-41,  150-52. 

Trabue,  M.  R.  "Completion-Test  Language  Scales."  Teachers  College  Contribu- 
tions to  Education,  No.  77.  New  York:  Teachers  College,  Columbia  University  1916, 
p.  29-73. 

Hughes,  J.  M.  "The  use  of  tests  in  the  evaluation  of  factors  which  condition  the 
achievement  of  pupils  in  high  school  physics,"  Journal  of  Educational  Psychology,  16: 
217-31,  April,  1925. 

[17] 


TABLE  II.     FINAL  VALUES  OF  ADDITION  EXERCISES 


No.  of 

Value 

No.  of 

Value 

No.  of 

Value 

No.  of 

Value 

Exercise 

Exercise 

Exercise 

Exercise 

2 

1.23 

14 

3.92 

22 

6.44 

35 

7.97 

3 

1.40 

9 

4.18 

19 

6.79 

29 

8.04 

5 

2.50 

12 

4.19 

23 

7.11 

31 

8.18 

7 

2.61 

13 

4.85 

34 

7.43 

24 

8.22 

6 

2.83 

15 

4.97 

26 

7.47 

36 

8.58 

8 

3.21 

17 

5.52 

30 

7.61 

37 

8.67 

1 

3.26 

16 

5.59 

27 

7.62 

33 

8.67 

4 

3.35 

18 

5.73 

25 

7.67 

38 

9.19 

10 

3.63 

20 

5.75 

28 

7.71 

11 

3.78 

21 

6.10 

32 

7.71 

below  the  average  and  another  25  percent  between  the  average  and 
1  Md.D.  above  it.  Furthermore  the  average  of  a  symmetrical  distribu- 
tion falls  at  the  middle  of  the  distribution,  so  that  50  percent  of  the 
scores  lie  below  it  and  50  percent  above  it.  Therefore,  it  is  easily  seen 
that  75  percent  of  the  scores  lie  below  1  Md.D.  above  the  average,  as 
this  is  simply  the  sum  of  the  50  percent  below  the  average  and  the  25  per- 
cent between  the  average  and  1  Md.D.  above  it.  To  make  this  clearer 
the  accompanying  figure  is  given.   The  portion  of  the  normal  frequency 


M.   -MMdLD. 


Figure  2.    Representation  of  a  Normal  Distribution  of  Scores 

Showing  the  Meaning  of  the  Median  Deviation  as 

a  Unit  of  Difficulty 


surface  to  the  left  of  the  vertical  line  at  its  center,  marked  M.s  is  the  50 
percent  of  the  area  below  the  average.  That  part  between  this  vertical 
fine  and  the  one  erected  at  -j-  1  Md.D.  is  the  25  percent  between  the 


[18] 


average  and  one  median  deviation  above  the  average.  Thus,  all  the 
area  to  the  left  of  the  shorter  vertical  line  is  75  percent  of  the  whole 
area.  With  this  in  mind  we  can  now  explain  the  meaning  of  the  median 
deviation  as  a  unit  of  difficulty  by  saying  that  it  is  the  difference  in 
difficulty  between  an  exercise  answered  correctly  by  50  percent  of  the 
pupils  tested  and  another  answered  correctly  by  75  percent  of  the 
pupils. 1S  Looking  at  Table  II  we  see  that  exercise  25  has  a  value  of 
7.67  and  exercise  37  of  8.67,  a  difference  of  1.00  Md.D.  ,.  We  know, 
therefore,  that  if  the  two  exercises  were  given  to  the  same  group  of 
pupils  and  50  percent  of  them  answered  exercise  37  correctly,  75  per- 
cent might  be  expected  to  answer  exercise  25   correctly,   since  it  is    1 

Md.D.     easier  than  the  former. 
u. 

There  is  also  another  somewhat  different  meaning  which  is  often 
attached  to  the  median  deviation  when  used  as  a  unit  of  measurement. 
In  the  construction  of  such  measuring  instruments  as  handwriting  and 
drawing  scales,  one  method  of  determining  the  value  or  merit  of  the 
specimens  being  rated  for  a  scale  is  to  have  them  compared  with  one 
another  by  a  number  of  supposedly  competent  judges.  For  example, 
judges  compare  specimen  A  with  B,  also  A  with  C,  B  with  C,  and  so  on. 
Record  is  made  of  how  many  or  what  percent  of  the  judges  rate  A  as 
better  than  B  and  of  course  how  many  rate  B  as  better  than  A,  and  so 
on.  When  75  percent  of  the  judges  rate  one  specimen  as  better  than 
another11'  the  difference  in  merit  between  the  two  is  assumed  to  be  1 
Md.D.  ,  .  This  is  illustrated  by  Figure  3  in  which  the  surfaces  under 
the  two  curves  are  assumed  to  represent  distributions  of  judges'  ratings 
of  two  specimens,  A  and  B.  It  is  assumed  that  the  opinions  of  judges 
concerning  the  merit  or  value  of  a  specimen  will  form  a  normal  distri- 
bution, the  center  or  average  of  which  is  the  true  value.  Therefore,  the 
surface  at  the  left,  under  curve  A,  is  taken  as  representing  the  distribu- 
tion of  judges'  opinions  concerning  specimen  A  and  the  point  A  on  the 
base  line  where  the  solid  vertical  line  meets  it  as  the  true  value  of  A. 
Similarly,  point  B  at  the  foot  of  the  broken  vertical  line  is  assumed  to 
represent  the  true  value  of  B.    If  75  percent  of  the  judges  rate  B  as 


lhIt  is  also  possible  to  say  that   1  Md.D.  _    is  the  difference  in  difficult)-  between 

an  exercise  answered  correctly  by  25  percent  of  the  pupils  and  one  answered  correctly 
by  SO  percent  of  them,  but  the  form  of  statement  given  above  is  more  usual. 

MIn  rating  specimens  for  the  purpose  being  discussed,  judges  are  expected  to  rate 
each  as  better  or  worse  than  each  of  those  with  which  it  is  compared.  If  they  rate  two 
as  equal,  the  rating  must  be  thrown  out  or  divided  between  the  two.  Therefore  if  75 
percent  of  judges  rate  one  specimen  as  better  than  another,  25  percent  must  rate  it 
as  worse. 

[19] 


Figure  3.    Illustration  of  Method  of  Determining  Difference  in 

Merit  of  Two  Specimens  by  Judges'  Ratings  of  One  as 

Better  or  Worse  Than  the  Other 


superior  to  A,  75  percent  of  the  area  of  the  surface  to  the  right,  repre- 
senting B,  will  lie  above  or  to  the  right  of  the  vertical  line  assumed  to 
show  the  true  value  of  A  and  of  course  25  percent  below  or  to  the  left 
of  that  line.  Since  50  percent  of  the  judges'  ratings  of  B  lie  above  its 
average  merit,  that  is,  to  the  right  of  the  broken  vertical  line  above 
point  B,  the  portion  of  the  surface  representing  B  which  is  included 
between  the  two  vertical  lines  must  be  75  percent  minus  50  percent,  or 
25  percent.  To  make  clear  which  this  is,  it  has  been  shaded  in  the  figure. 
We  have  already  seen  that  a  distance  of  1  median  deviation  in  one 
direction  from  the  average  distribution  includes  25  percent  of  the  total 
number  of  cases.  Therefore,  the  distance  between  the  two  vertical  lines 
must  be  1  Md.D.  in  order  that  25  percent  of  the  area  be  included. 

This  method  of  determining  the  value  of  merit  of  specimens  has 
been  made  use  of  in  the  case  of  a  number  of  our  standardized  scales. 
Probably  the  best  known  example  of  its  use  is  in  connection  with  Thorn- 
dike's  Handwriting  Scale.-"  In  his  account  of  its  construction  he  describes 
two  methods,  one  of  which  is  that  just  mentioned.  He  had  samples  of 
handwriting  rated  by  a  number  of  judges  as  to  whether  they  were  better 


2°Thorndike,  E.  L.  '"Handwriting,"  Teachers  College  Record,  11:1-41,  March, 
1910.    For  further  examples,  see: 

Hoke,  E.  R.  "The  Measurement  of  Achievement  in  Shorthand."  The  Johns  Hop- 
kins University  Studies  in  Education,  No.  6.  Baltimore:  Johns  Hopkins  Press,  1922, 
p.  33-34. 

Hillegas,  M.  B.  '"Scale  for  the  measurement  of  quality  in  English  composition 
by  young  people,"  Teachers  College  Record,  13:1-54,  September,  1912. 

Murdoch,  Katherine.  "'The  Measurement  of  Certain  Elements  of  Hand  Sewing." 
Teachers  College  Contributions  to  Education,  No.  103.  New  York:  Teachers  College, 
Columbia  University,  1919.  p.  22-26. 


[20] 


or  worse  than  the  other  samples  and,  according  to  the  method  outlined 
above,  determined  the  differences  in  merit  between  the  samples  in  terms 
of  the  median  deviation.  A  sample  considered  to  possess  no  merit  as 
handwriting,  though  obviously  an  attempt  to  write,  was  used  as  the  zero 
point  and  the  distance  of  each  sample  above  this  point  determined. 

Probable  errors  of  sampling.  The  third  use  of  the  probable  error 
is  one  to  which  that  term  is  properly  applied.  In  this  case  it  is  employed 
directly  as  a  measure  of  the  size  of  the  errors  involved  in  sampling,  that 
is  to  say,  as  a  measure  of  the  reliability  of  sampling.  The  probable 
error  of  sampling  can  not  be  used  alone,  but  must  always  be  connected 
with  some  other  measure  such  as  an  average,  a  standard  or  quartile 
deviation,  a  difference,  a  coefficient  or  ratio  of  correlation,  a  regression 
coefficient,  or  other  similar  measures.  Assuming  that  the  sample  has 
been  selected  in  a  random  manner,  in  other  words  that  it  is  not  biased, 
the  probable  error  of  sampling  gives  an  indication  of  how  reliable  such 
derived  measures  are  when  the  cases  upon  which  they  are  based  are 
considered  as  a  sample  of  a  larger  number  of  similar  ones.  For  example, 
if  the  average  score  of  five  hundred  eighth-grade  children  upon  an  intel- 
ligence test  has  been  determined  and  it  is  assumed  that  no  errors  are 
present  in  the  test  scores  or  computations  leading  to  the  average,  this 
average  is  the  true  one  for  the  children  actually  tested.  If  the  five  hun- 
dred children  have  been  selected  from  a  much  larger  number  in  a  city 
school  system,  the  average  obtained  from  their  scores  is  not,  except  by 
chance,  the  true  average  of  all  the  eighth-grade  children  in  the  system. 
However,  if  we  assume  that  the  five  hundred  children  constitute  a  ran- 
dom sample,  we  can  determine  the  reliability  of  the  average  actually 
obtained  when  considered  as  the  average  of  all  of  the  eighth  grade 
children  in  the  system. 

When  the  probable  error  of  sampling  is  used,  it  is  both  customary 
and  convenient  to  place  a  plus  and  minus  sign,  followed  by  the  probable 
error,  immediately  after  the  measure  to  which  it  applies.  Thus  if  the 
average  intelligence  quotient  of  the  five  hundred  pupils  had  been  found 
to  be  102  and  its  probable  error  3,  it  would  frequently  be  written 
102  ±  3,  when  considered  as  an  average  I.O.  of  all  of  the  eighth-grade 
pupils  in  the  system.  The  same  practice  is  also  followed  in  the  case  of 
other  measures  than  the  average.  A  second  fairly  common  way  of  re- 
ferring to  the  probable  error  of  sampling  is  to  use  the  abbreviation  P.E. 
with  a  subscript  indicating  the  measure  to  which  it  applies.  Thus  P.E. 


M 


denotes  the  probable  error  of  the  mean,  P.E.       that  of  the  median,  P.E. 
that  of  the  coefficient  of  correlation,  and  so  on. 

[21] 


The  interpretation  of  the  probable  error  of  sampling  from  the 
standpoint  of  chance  is  the  same  as  that  of  the  median  deviation  when 
used  as  a  measure  of  variability  or  scatter.  That  is,  the  chances  are 
even  that  the  true  measure  of  the  whole  group  does  not  differ  from  the 
measure  obtained  from  the  sample  by  more  than  the  value  of  the 
probable  error;  they  are  4.6  to  1  that  it  does  not  differ  by  more  than 
2  P.E.,  22  to  1  that  it  does  not  differ  by  more  than  3  P.E.,  and  so  on. 
Another  way  of  stating  the  same  thing  is  that  if  a  number  of  samples 
of  the  same  size  as  the  one  already  taken  and  similar  to  it  were  selected 
and  corresponding  measures  computed  from  them,  half  of  these  measures 
would  probably  fall  within  1  P.E.  of  the  first  one  computed,  82  percent 
within  2  P.E.,  96  percent  within  3  P.E.,  and  so  on.  Thus,  in  the  case  of 
the  group  of  eighth-grade  pupils  referred  to.  it  is  probable  that,  if  a 
number  of  similar  samples  were  chosen  and  their  means  determined, 
half  of  them  would  fall  within  3  points  of  102,  that  is  between  99  and 
105,  82  percent  between  96  and  108  (102  ±6),  96  percent  between  93 
and  111  (102  ±9),  and  so  forth. 

A  good  example  of  this  use  of  the  probable  error  is  to  be  found  in 
a  recent  issue  of  the  Journal  of  Educational  Psychology.-1  In  the  article 
referred  to  the  following  table  is  given.  It  contains  a  number  of  means, 
standard  deviations,  and  coefficients  of  correlation,  each  followed  by  its 
probable  error.  For  example,  the  mean  English  grade  of  the  first  high- 
school  group  is  given  as  84.6  ±  .3.  This  indicates  that  if  similar  sam- 
ples were  taken  it  is  probable  that  half  of  the  obtained  means  would  be 
between  84.3  and  84.9  (84.6  ±  .3),  82  percent  of  them  between  84.0 
and  85.2  (84.6  ±  .6).  96  percent  between  83.7  and  85.5  (84.6  ±  .9), 
and  so  on. 

The  formulae  by  which  to  compute  a  probable  error  of  sampling 
differ  according  to  the  measure  for  which  it  is  being  found.   The  follow- 


21Gowex,  Johx  W.,  and  Gooch,  Marjorie.  "The  mental  attainments  of  college 
students  in  relation  to  previous  training,"  Journal  of  Educational  Psychology,  16:547-68. 
November,  1925.   Other  examples  may  be  found  in  the  following  references: 

Rich,  S.  G..  and  Skixxer,  C.  E.  "Intelligence  among  normal  school  students." 
Educational  Administration  and  Supervision,  11:639-44,  December,  1925. 

Ellis,  R.  S.  "A  comparison  of  the  scores  of  college  freshmen  and  seniors  on  psy- 
chological tests,"  School  and  Society.  23:310-12.  March  6,  1926. 

Remmers,  H.  H.,  and  Edxa  M.  "The  negative  suggestion  effect  of  true-false  exam- 
ination questions."  Journal  of  Educational  Psychology,  17:52-56,  January.   1926. 

Moxroe,  \V.  S.  An  Introduction  to  the  Theory  of  Educational  Measurements. 
Boston:  Houghton  Mifflin  Company,  1923,  p.  204. 

[22] 


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[23] 


ing  are  the  formulae  for  the  probable  errors  of  the  mean,  the  median, 
the  standard  deviation,  and  the  coefficient  of  correlation: 

Md.B. 

jr.  r..yi   —         ^^ 


P.KMim-  1-2533^ 
P.  E.    =  .7071  Md-  D- 


P.E.    = 


Vn 

1  -  r- 

vw 


In  these  typical  formulae  it  will  be  noticed  that  there  is  one  common 
element, ^/Ar,  appearing  in  their  denominators.  The  same  is  true  of  the 
formulae  for  the  probable  errors  of  sampling  of  almost  all  commonly 
used  statistical  measures.  Even  in  those  cases  in  which  -y/^ydoes  not 
appear  directly  in  the  denominator  of  the  formulae,  it  or  some  similar 
expression  is  usually  in  some  way  contained  therein.  Since  N  stands 
for  the  number  of  cases  in  the  sample,  it  can  easily  be  seen  that  the 
larger  the  sample  the  larger  is  the  denominator  of  the  fraction  and 
therefore  the  smaller  the  value  of  the  probable  error.  In  other  words, 
increasing  the  size  of  the  sample  decreases  the  size  of  the  probable 
errors  present  and  hence  increases  the  reliability  or  accuracy  of  the 
derived  measures. 

The  probable  error  of  a  number  of  measurements  of  the  same 
thing.  Another  situation  in  which  the  term  probable  error  is  appropri- 
ate is  in  measuring  the  size  of  the  variable  errors  present  in  a  number 
of  measurements  of  the  same  thing.  In  most,  if  not  all,  situations  it  is 
impossible  to  measure  a  trait  with  such  a  high  degree  of  precision  and 
reliability  that  all  similar  measurements  will  agree  exactly  with  the 
original  one.  For  example,  let  us  suppose  that  ten  different  persons 
determine  a  child's  height  or  that  the  same  person  does  so  ten  times. 
If  height  is  being  found  only  to  the  nearest  inch  and  the  persons  doing 
the  measuring  are  fairly  competent  it  is  likely  that  all  the  results  will 
agree.  If,  however,  the  attempt  is  made  to  secure  a  rather  high  degree 
of  accuracy  and  results  are  given,  let  us  say,  to  the  nearest  sixteenth  of 
an  inch,  it  is  extremely  improbable  that  the  results  obtained,  whether 
bv  ten  different  persons  or  by  the  same  person  at  ten  different  times, 
will  be  the  same.  There  are  generally  two  causes  for  this  and  fre- 
quently a  third  one.   In  the  first  place,  even  though  the  persons  making 

[24] 


the  measurements  are  reasonably  competent  it  is  unlikely  that  all  have 
just  the  qualities,  such  as  keenness  of  eyesight,  steadiness  of  hand,  abil- 
ity to  time  accurately,  and  so  forth,  necessary  for  accurate  measure- 
ment or  that  all  exercise  exactly  the  same  degree  of  care.  Secondly,  it 
is  improbable  that  the  child  being  measured  will  assume  exactly  the 
same  posture  when  all  ten  measurements  are  being  made.  In  addition, 
if  different  measuring  instruments  are  used  it  is  very  unlikely  that  they 
are  absolutely  identical.  The  errors  due  to  all  these  and  any  other 
chance  causes  are  called  variable  errors2-  and  are  often  measured  by 
the  probable  error.  In  a  sense  they  may  be  thought  of  as  errors  of 
sampling,  for  just  as  a  group  too  large  to  have  all  its  members  meas- 
ured is  sampled  by  measuring  a  part  of  them,  so  a  characteristic  which 
cannot  be  measured  with  absolute  accuracy  and  therefore  theoretically 
requires  that  an  infinite  number  of  measurements  be  made  and  averaged 
to  secure  a  perfect  one,  is  sampled  by  making  a  limited  number  of 
measurements. 

A  common  example  of  the  occurrence  of  variable  errors  is  in  con- 
nection with  the  giving  of  written  examinations  and  tests.  At  one  test- 
ing period  a  pupil  may  happen  to  be  feeling  unusually  well,  whereas  at 
another  his  health  may  be  below  par;  at  one  time  he  may  have  re- 
viewed the  material  covered  by  the  questions  recently,  but  at  another 
it  may  happen  that  the  questions  touch  material  about  which  he  knows 
little  although  he  remembers  most  of  what  he  has  studied;  at  one  time 
he  may  make  a  better  score  than  he  deserves  by  cheating,  whereas  at 
another  his  score  may  not  indicate  his  true  ability  because  his  pencil 
broke  or  something  outside  the  window  attracted  his  attention,  and  so 
on.  Similarly,  when  weight  is  being  measured  the  result  will  vary  ac- 
cording to  whether  or  not  the  individual  has  eaten  a  meal  recently, 
whether  he  is  wearing  heavier  or  lighter  clothing  than  usual,  has  more 
or  less  in  his  pockets,  and  so  on. 

Since,  because  of  all  these  variable  errors,  we  can  rarely,  if  ever, 
establish  that  any  one  obtained  score  is  a  true  or  even  the  best  obtain- 
able measurement  of  the  characteristic  being  dealt  with,  the  best  that 
we  can  do  is  to  supplement  the  scores  obtained  by  a  statement  of  their 
reliability.  As  in  the  case  of  the  probable  error  of  sampling  so  here  the 
P.E.  is  commonly  affixed  to  the  obtained  measure  with  a  plus  or  minus 
sign   connecting  the   two.     Thus,   a   pupil's    height  may   be   stated   as 


^For  a  fuller  discussion  of  variable  errors  see: 

Monroe,  W.  S.  "The  constant  and  variable  errors  of  educational  measurements." 
University  of  Illinois  Bulletin,  Vol.  21,  No.  10,  Bureau  of  Educational  Research  Bulletin 
No.  15.   Urbana:    University  of  Illinois,  1923.    30  p. 

[25] 


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[26] 


62.00  ±  .25  inches,  which,  of  course,  means  that  if  it  is  measured  a 
number  of  times  half  of  the  measurements  will  probably  fall  between 
61.75  and  62.25  inches,  82  percent  of  them  between  61.50  and  62.50 
inches,  and  so  on.  An  example  of  this  use  of  the  probable  error  (P.E.) 
is  shown  in  Table  IV",  taken  from  Kelly, ";:  who  made  use  of  data  col- 
lected by  Starch  and  Elliott.24  This  table  shows  the  marks  given  to 
each  of  three  papers  by  several  groups  of  teachers.  The  first  column, 
for  example,  shows  that  of  ninety-one  teachers  in  schools  with  passing 
marks  of  75  two  rated  paper  A  from  60  to  64,  inclusive,  one  rated  it 
between  65  and  69,  two  between  70  and  74,  and  so  on.  The  median 
rating  was  88.1  and  the  probable  error  of  the  ratings  4.9.  In  other 
words,  half  of  them  were  within  4.9  points  of  88.1  and  half  were  not. 
To  make  the  situation  more  concrete  Figure  4  has  been  prepared. 


No.of 

§£ses 


25 


ZO 


15 


10 


5 


0i—  --ra 


25        35 


%S 


15        55       65       IS 
Percent    Marks 

Figure  4.    Graphical  Representation  of  Marks  Given  in  the 
Last  Column  of  Table  IV 


"Kelly,  F.  J.  "Teachers  Marks:  Their  Variability  and  Standardization."  Teach- 
ers College  Contributions  to  Education  No.  66.  New  York:  Teachers  College,  Colum- 
bia University,  1914,  p.  55.    Also  see: 

Trabue,  M.  R.  Measuring  Results  in  Education.  New  York:  American  Book 
Company,  1924,  p.  199-203,  259-67. 

"Starch,  D.,  and  Elliott,  E.  C.  "Reliability  of  the  grading  of  high  school  work 
in  English,"  School  Review,  20:442-57,  September,  1912. 

Starch,  D.,  and  Elliott,  E.  C.  "Reliability  of  grading  work  in  mathematics," 
School  Review,  21:254-59,  April,  1913. 


[27] 


It  represents  graphically  the  distribution  of  marks  shown  in  the  last 
column  of  Table  IV.  At  a  distance  of  1  P.E.,  that  is  8.0,  from  the 
median,  which  is  70.3,  vertical  lines  have  been  erected.  It  can  be  seen 
by  rough  inspection  that  approximately  half  of  the  area  of  the  column 
diagram  lies  between  these  two  lines  or,  in  other  words,  half  of  the 
marks  fall  within  these  two  lines  and,  of  course,  the  other  half  outside 
of  them. 

The  best  estimate  of  the  true  rating  of  the  paper  is  70.3,  the 
median,  with  a  P.E.  of  8.0..  which  means  that  half  of  the  marks  given 
by  the  group  of  116  persons  who  rated  the  paper  probably  fall  within 
8.0  points  of  70.3,  that  is,  between  62.3  and  78.3,  82  percent  of  them 
fall  between  54.3  and  86.3,  and  so  on.  The  same  may  also  be  expected 
of  marks  given  by  other  raters  equally  competent  with  those  in  the  first 
group. 

Errors  of  estimate  and  measurement.  Since  none  of  our  standard- 
ized tests  and  scales  of  other  measuring  instruments  are  perfectly  reli- 
able, that  is,  since  two  or  more  applications  of  the  same  instrument  or 
of  supposedly  equivalent  forms  thereof  cannot  be  relied  upon  to  yield 
exactly  the  same  measurements,  there  are  evidently  some  errors  in- 
volved. When  the  data  consist  of  two  series  of  scores  or  measurements 
of  a  number  of  individuals25  rather  than  of  a  number  of  measurements 
of  one  individual,  the  errors  involved  are  known  as  errors  of  estimate 
and  of  measurement.  Since  these  tend  to  form  normal  distributions,  as 
do  all  other  variable  errors,  the  median  deviation  of  their  distribution 
may  be  used  as  a  measure  of  their  magnitude.  When  this  is  done  the 
terms  probable  error  of  estimate  and  probable  error  of  measurement 
are  applied.     These  are  commonlv  abbreviated  P.E.  „     and  P.E.  , 

r  r  J  Est.  Meas. 

Occasionally,    instead    of    P.E.        one    finds    P.E.,.        and    instead    of 

*  Est.  Score 

P.E.        ,  P.E*  ,  .     The  writer  recommends,  however,  that  these  latter 

Meas.  M. 

abbreviations  not  be  used  since  they  might  be  confused  with  those  for 
other  uses  of  the  probable  error. 


25That  is,  when  each  individual  has  been  measured  twice.  The  first  measurements 
of  all  individuals  constitute  one  series  and  the  second  ones  the  other.  Such  data  as  these 
or  any  other  which  are  correlated  are  frequently  called  ''sets  of  paired  facts." 


[28] 


The  most  commonly  used  formula26  for  these  measures  are  as  fol- 
lows : 

P.  E.EsU  =  .6745  a Vl  -  r\2  or  Md.  D.  y/1  -  r[2  and 


P-  E-Mtas.  =  -6745        -       VI  -  ru  or-  0  — -Vl  - 


r 


In  the  first,  the  a  or  Md.D.  used  is  that  of  the  distribution  of  which  the 
scores  are  being  estimated.  That  is.  if  scores  in  the  second  series  are 
being  estimated  from  those  in  the  first,  the  a  or  Md.D.  of  the  second  is 
used.  In  the  second,  the  two  o-'s  or  Md.D.'s  are  averaged.  The  r12  in 
both  is  the  coefficient  of  correlation  between  the  two  series  of  scores. 

As  was  suggested  above,  the  probable  error  of  estimate  is  a  meas- 
ure of  the  differences  between  the  results  obtained  by  measuring  a  group 
of  individuals  a  first  and  a  second  time  with  the  same  or  similar  instru- 
ments. Occasionally,  this  definition  is  extended  to  include  the  differ- 
ences between  any  two  series  of  scores  of  the  same  individuals  if  they 
are  used  for  purposes  of  predicting  or  estimating  one  in  terms  of  the 
other.  The  probable  error  of  measurement  differs  in  that  it  measures 
the  differences  between  the  scores  obtained  from  one  of  the  two  series 
of  measurements  and  the  theoretically  true  scores  of  the  individuals 
tested.  Since  the  theoretically  true  scores  are  the  averages  of  infinite 
numbers  of  obtained  scores  with  practice  effects  and  all  other  constant 
errors  eliminated,  it  is  impossible  to  determine  them.  If  two  series  of 
scores  are  available,  however,  it  is  possible  to  compute  measures  of  the 
size  of  the  differences  between  a  series  of  actually  obtained  scores  and 
the  theoretically  true  ones.  The  probable  error  of  estimate  is  always 
larger  than  the  probable  error  of  measurement  because  the  differences 
between  two  series  of  scores,  both  of  which  contain  variable  errors,  are 
naturally  greater  than  those  between  a  series  of  scores  containing  vari- 
able errors  and  another  series  of  theoretically  true  scores  which  contain 
no  such  errors. 

Table  V,  taken  from  a  critical  study  of  a  standardized  test,27  gives 
certain  results  obtained  from  giving  two  intelligence  scales,  the  Illinois 
and  the  National,  to  several  thousand  pupils.     The  number  of  pupils 


^Several  other  formulae  are  sometimes  employed,  especially  for  the  probable  error 
of  measurement.    See: 

Odell,  op.  cit..  p.  230-41. 

2TMonroe,  Walter  S.  "The  Illinois  Examination."  University  of  Illinois  Bulletin, 
Vol.  19,  No.  6,  Bureau  of  Educational  Research  Bulletin  No.  6.  Urbana:  University  of 
Illinois,  1921,  p.  58. 

[29] 


TABLE  V.    CORRELATION  BETWEEN  SCORES  YIELDED  BY  ILLINOIS 

GENERAL  INTELLIGENCE  SCALE  AND  BY  NATIONAL 

INTELLIGENCE  SCALE 


Number  of 

P.  E.Est 

Grade 

Cases 

r 

P.  E.Est. 

Average 

III  A 

357 

0.53 

9.1 

0.22 

IV  B 

416 

0.70 

9.6 

0.18 

IV  A 

335 

0.74 

8.0 

0.14 

V  B 

460 

0.55 

8.7 

0.14 

V  A 

285 

0.47 

12.0 

0.19 

VI  B 

383 

0.44 

12.6 

0.17 

VI  A 

259 

0.67 

10.8 

0.13 

VII  B 

350 

0.70 

11.0 

0.12 

VII  A 

210 

0.68 

10.3 

0.11 

VIII  B 

271 

0.72 

10.2 

0.10 

VIII  A 

289 

0.69 

10.9 

0.10 

All  Grades 

3615 

0.81 

11.5 

0.16 

tested,  the  coefficient  of  correlation,  the  probable  error  of  estimate  and 
the  ratio  of  P.E.  „     to  the  average  are  given  for  each  half-grade  from 

Est.  " 

IIIA  to  VIIIA,  inclusive,  also  for  all  combined.  The  meaning  of  the 
P.E.  in  the  last  line,  for  example,  is  that  if  probable  scores  of  pupils 
in  Grades  IIIA  to  VIIIA  upon  the  Illinois  General  Intelligence  Scale 
are  estimated  from  actual  scores  upon  the  National  Intelligence  Scale, 
half  of  them  will  be  in  error  by  less  than  11.5  points  and  half  of  them 
by  more  than  that  amount.  Furthermore,  82  percent  will  be  in  error 
by  less  than  twice  11.5  or  23,  96  percent  by  less  than  34.5,  and  so  on. 
An  example  of  the  use  of  the  probable  error  of  measurement  may 
also  be  cited  from  the  same  source.2S    Table  VI  gives,  for  Grades  III 


^Monroe,  op.  cit.,  p.  49.  Other  illustrations  may  be  found  by  consulting  the  fol- 
lowing sources: 

Monroe,  Walter  S.  "A  Critical  Study  of  Certain  Silent  Reading  Tests."  Uni- 
versity of  Illinois  Bulletin,  Vol.  19,  No.  22,  Bureau  of  Educational  Research  Bulletin 
No.  8.   Urbana:    University  of  Illinois,  1922,  p.  33-34. 

Monroe,  W.  S.,  Devoss,  J.  C,  and  Kelly,  F.  J.  Educational  Tests  and  Measure- 
ments, Revised  and  Enlarged.    Boston:  Houghton  MifHin  Company,  1924,  p.  410. 

Thorndike,  E.  L.,  and  Symonds,  P.  M.  "Difficulty,  reliability,  and  grade  achieve- 
ments in  a  test  of  English  vocabulary,"  Teachers  College  Record,  34:438-45,  Novem- 
ber, 1923. 

Dearborn,  W.  F.  "Reliability  and  uses  of  group  tests  of  intelligence."  Eleventh 
Conference  on  Educational  Measurements.  Bulletin  of  the  School  of  Education,  Indiana 
University,  Vol.  1,  No.  3.   Bloomington:  Indiana  University,  1925,  p.  115-30. 


[30] 


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[31] 


to  VIII,  the  probable  errors  of  measurement  of  the  three  tests  which 
make  up  the  Illinois  Examination,  also  their  ratios  to  the  averages.  Tak- 
ing the  entries  in  the  first  line  of  the  table  as  an  illustration,  half  of  the 
differences  between  the  intelligence  scores  actually  obtained  in  Grade 
III  with  forms  1  and  2  and  the  theoretically  true  scores  were  found  to 
be  less  than  3.5  points  and  half  of  them  greater  than  this  amount.  For 
the  arithmetic  test  half  of  these  differences  were  less  than  2.6  points; 
for  the  comprehension  scores  of  the  silent  reading  test  half  of  the  dif- 
ferences were  less  than  1.2,  and  for  the  rate  scores  half  of  the  differences 
less  than  13.7  words  per  minute.  The  probable  errors  of  estimate, 
which  are  not  given  in  Table  VI,  would  of  course  be  larger,  that  cor- 
responding to  the  probable  error  of  measurement  of  13.7  just  mentioned 
being  about  18.5  words  per  minute. 

It  will  be  noticed  that  in  Tables  V  and  VI  the  columns  containing 
the  probable  errors  of  estimate  and  of  measurement,  respectively,  are 
followed  by  columns  showing  the  ratios  of  these  measures  to  the  corre- 
sponding averages.20  This  is  done  because  the  mere  statement  of  the 
size  of  a  probable  error  of  estimate  or  of  measurement  usually  conveys 
little  definite  meaning  unless  one  knows  the  size  of  the  individual  meas- 
ures themselves.  Just  as  an  error  of  an  inch  is  of  slight  significance  in 
measuring  the  distance  between  two  cities  or  even  the  length  of  a  lot 
but  is  relatively  significant  in  measuring  a  person's  height  and  very  im- 
portant in  fitting  a  piston  to  its  cylinder,  so  an  error  of  a  given  number 
of  points  on  a  test  becomes  more  significant  the  smaller  the  score.  It 
will  be  seen  that  whereas  Tables  V  and  VI  show  either  a  slight  tendency 
for  the  probable  errors  to  be  greater  in  the  higher  grades  or  else  no 
regular  tendency  at  all,  they  reveal  that  relative  to  the  average  scores 
which  increase  from  grade  to  grade,  the  errors  become  smaller,  the  ratios 
being  considerably  less  in  the  eighth  grade  than  in  the  third. 


29There  are  certain  objections  raised  to  the  use  of  these  ratios  which  will  not  be 
discussed  here,  further  than  to  admit  that  sometimes  their  use  may  be  misleading.  The 
writer  believes,  however,  that  in  general  their  use  is  desirable  both  because  the  probable 
errors  alone  frequently  convey  little  helpful  information  and  because  no  better  relative 
measure  has  been  suggested. 

[32] 


CHAPTER  III 
THE  COEFFICIENT  OF  CORRELATION 

Definition  of  correlation.  Before  proceeding  to  discuss  the  use  and 
interpretation  of  the  coefficient  of  correlation  it  seems  in  order  to  define, 
first,  what  is  meant  by  correlation  in  general,  and,  second,  what  is 
meant  by  the  coefficient  of  correlation.  Two  characteristics  or  traits  are 
said  to  be  correlated  when  there  is  a  tendency  for  changes  in  the  value 
of  one  to  be  associated  or  occur  concurrently  with  changes  in  the  value 
of  the  other.  If  most  of  the  changes  in  one  of  the  things  being  dealt 
with  are  in  the  same  direction  as  the  corresponding  changes  in  the  other, 
the  correlation  is  said  to  be  positive  or  direct;  if  in  opposite  directions, 
it  is  said  to  be  negative  or  inverse.  For  example,  if  pupils'  marks  in  al- 
gebra and  English  are  being  correlated,  and  in  most  cases  pupils  who 
are  relatively  high  in  one  are  also  relatively  high  in  the  other  and  like- 
wise those  who  are  low  in  one  are  generally  low  in  the  other,  the  corre- 
lation is  positive;  whereas,  if  pupils  who  stand  high  in  algebra  tend  to 
rank  low  in  English,  and  vice  versa,  it  is  negative.  The  greater  the 
proportion  of  associated  changes  which  are  in  the  same  direction,  the 
greater  is  the  amount  of  positive  correlation;  the  greater  the  proportion 
in  opposite  directions,  the  greater  the  negative  correlation.  It  is  also 
true  that  the  greater  the  agreement  in  relative  magnitude  of  the  con- 
current changes,  the  greater  the  degree  of  correlation,  whether  positive 
or  negative.  For  example,  if  a  pupil  who  is  10  percent  above  the  aver- 
age in  English  is  also  10  percent  above  the  average  in  algebra,  if  one 
who  is  5  percent  above  in  English  is  5  percent  above  in  algebra,  and  so 
on  for  most  of  the  cases,  the  correlation  is  higher  than  if  this  condition 
does  not  obtain. 

Examples  of  both  positive  and  negative  correlation  are  very  numer- 
ous and  easily  found.  For  example,  it  is  usually  found  that  the  greater 
a  person's  height,  the  greater  his  weight;  and  that  the  older  a  child,  the 
greater  his  strength.  Therefore,  height  and  weight,  and  children's  age 
and  strength  are  positively  correlated.  On  the  other  hand,  after  an 
adult  passes  a  certain  age  strength  tends  to  decrease  with  advancing 
years  so  that  the  correlation  is  negative.  This  is  also  true  when  the  two 
things  compared  are  size  of  class  and  cost  of  instruction  per  pupil,  since, 
on  the  whole,  the  larger  the  class  the  smaller  is  the  cost  for  each  mem- 
ber thereof. 

[33] 


The  fact  should  be  emphasized  that  the  existence  of  correlation 
does  not  prove  that  there  is  any  dependence  or  causal  relationship  be- 
tween the  two  things  correlated.  It  may  be  that  such  dependence  exists, 
but  it  may  also  be  that  neither  trait  in  any  sense  causes  the  other.  In- 
stead, the  existing  correlation  may  be  due  to  the  action  of  one  or  more 
outside  factors  which  affect  both  the  characteristics  being  dealt  with. 
Sometimes  the  causal  factor  or  factors  may  be  even  more  remote  than 
this,  that  is,  some  common  cause  may  affect  two  characteristics  or  fact- 
ors, each  of  these  two  may  affect  another,  and  so  on,  with  the  result 
that  the  final  pair  of  characteristics  considered,  though  relatively  remote 
from  the  common  cause,  show  correlation  with  each  other.  On  the 
other  hand,  if  the  correlation  between  two  traits  is  fairly  high,  the  like- 
lihood that  one  of  them  affects  the  other  or  that  both  are  affected  by  a 
relatively  proximate  common  cause,  is  great  enough  to  be  investigated 
as  a  probable  hypothesis. 

Definition  of  the  coefficient  of  correlation.  Although  "coefficient 
of  correlation"  is  sometimes  used  in  a  broad  sense  toinclude  any  one  or  all 
of  a  number  of  numerical  expressions  which  summarize  the  degree  of 
relationship  between  two  variables,  it  is  best  to  reserve  this  term  for  the 
product-moment  coefficient  of  correlation,  sometimes  called  the  Pear- 
son coefficient  because  its  present  extensive  use  is  chiefly  due  to  the 
English  statistician,  Karl  Pearson.  This  expression,  which  is  abbrevi- 
ated by  "r",  is  given  by  the  formula: 

2xy 


Na    <r 

x     y 


in  which  x  and  y  represent  the  deviations  of  the  individual  measures 

from  their  respective  averages,  and  a      and  a      the  standard  deviations 

x  y 

of  the  distributions  of  the  two  variables.  2  is  the  sign  of  summation 
and  N  stands  for  the  number  of  individuals.  Therefore,  what  the  for- 
mula accomplishes  is  to  multiply  the  deviations  or  differences  of  each 
case  from  the  averages,  find  the  sum  of  these  products  for  all  the 
cases  concerned,  divide  by  the  number  of  cases  to  find  the  average  pro- 
duct, and  further  divide  by  the  product  of  the  two  standard  deviations 
in  order  to  reduce  the  two  distributions  to  a  unit  which  is  common  and 
such  that  the  value  of  the  result  cannot  be  greater  than  ±  1.00. 

The  coefficient  of  correlation  is  a  measure  of  rectilinear  or  straight- 
line  relationship  only.  That  is,  it  measures  the  degree  to  which  the 
data  when  tabulated  in  a  correlation  table  approach  a  straight  line  or, 
in  other  words,  the  degree  to  which  their  graphical  representation  upon 

[34] 


TABLE  VII.   AN  EXAMPLE  OF  A  HIGH  COEFFICIENT  OF  CORRELATION: 

THE  CORRELATION  BETWEEN  POINT  SCORES  ON  THE  OTIS 

SELF-ADMINISTERING  TEST,  HIGHER  EXAMINATION, 

FORM  A  AND  INTELLIGENCE  QUOTIENTS  FOR 

A    GROUP   OF   HIGH    SCHOOL   SENIORS 


Intell 

igence  Quotients. 

Point 

Scores 

T 

61- 

-  66- 

71-  76 

-  8 

I-  86-  91 

-  96- 

101- 

106-111-116-1 

21-1 

26-131- 

71- 

1 

9 

10 

66- 

1 

12 

39 

2 

54 

61- 

1  30 

89 

9 

129 

56- 

1  47  167 

15 

2 

232 

51- 

74  264  14 

352 

46- 

91 

298  11 

400 

41- 

87 

245 

8 

340 

36- 

59 

180 

2 

241 

31- 

1 

23 

79 

4 

107 

26- 

11 

38 

1 

1 

51 

21- 

5 

14 

19 

16- 

1 

2 

3 

11- 

2 

r  =   .97 

2 

6- 

1- 

1 

1 

T 

1 

3 

7 

26 

61  1 

39 

272 

338 

381  323  212 

116 

51 

11 

1941 

the  X-  and  Y-  axes1  approaches  a  straight  line.  Tables  VII  and  VIII, 
taken  from  an  unpublished  study  made  by  the  writer,  are  inserted  to 
illustrate,  respectively,  rather  close  and  very  little  approach  to  straight- 
line  correlation.  In  Table  VII  the  intelligence  quotients  of  a  number  of 
high-school  seniors  are  shown  upon  the  X-  or  horizontal  axis,  and  the 
point  scores  of  the  same  seniors  upon  the  Y-  or  vertical  axis.  This 
table  shows,  taking  the  top  row  as  an  example,  that  one  senior  who  had 
a  point  score  between  71  and  75,  inclusive,  had  an  intelligence  quotient 
between  126  and  130,  inclusive,  and  that  nine  seniors  whose  point  scores 
were  from  71  to  75  had  intelligence  quotients  from  131  to  135.  It  can 
be  seen  that  if  a  straight  line  were  drawn  diagonally  through  the  table 
from  the  lower  left-hand  to  the  upper  right-hand  corner,  the  departure 
of  the  entries  from  it  would  be  comparatively  small.  In  other  words 
the  coefficient  of  correlation  of  .97  indicates  rather  close  approach  to 
perfect  straight-line  relationship.  In  Table  VIII  the  horizontal  or  X- 
axis  represents  the  number  of  semesters  of  Latin  carried  in  high  school, 


Tor  further  explanation  of  the  A*-  and  }'-  axes  see: 
Odell,  op.  cit.,  p.  37-38. 


[35] 


TABLE  VIII.    AN  EXAMPLE  OF  A  RATHER  LOW  COEFFICIENT  OF  COR- 
RELATION:    THE  CORRELATION  BETWEEN  FRESHMAN 
LATIN  MARKS  AND  NUMBER  OF  SEMESTERS  OF  LATIN 
CARRIED  IN  HIGH  SCHOOL  FOR  A  GROUP  OF 
COLLEGE  FRESHMEN 


Freshman 

Semesters 

of  L 

atin  in  High 

School 

Latin 

T 

Mark 

0 

1            2 

3 

4 

5 

6 

y 

8 

96- 

1 

3 

1 

l 

4 

10 

91- 

1 

3 

4 

i 

8 

17 

86- 

2 

1 

12 

3 

i 

17 

37 

81- 

5 

1 

8 

1 

i 

12 

28 

76- 

2 

3 

6 

1 

2 

4 

18 

71- 

1 

2 

1 

1 

i 

3 

9 

66- 

2 

2 

4 

61- 

1 

r 

=  .31 

1 

56- 

3 

1 

1 

1 

6 

T 

16 

6 

1 

38 

3 

11 

6 

49 

130 

and  the  vertical  or  Y-  axis  the  freshman  Latin  mark  in  college.  It 
shows,  for  example,  that  of  the  students  whose  freshman  Latin  marks 
were  from  96  to  100,  inclusive,  one  had  carried  no  Latin  in  high  school, 
three  had  carried  four  semesters,  one  five  semesters,  one  seven  semes- 
ters, and  four  ten  semesters.  An  inspection  of  the  table  shows  that,  as 
is  indicated  by  the  coefficient  of  only  .31,  it  is  impossible  to  draw  a 
straight  line  through  it  in  such  a  direction  that  the  entries  show  any 
considerable  tendency  to  lie  near  this  line. 

It  should  also  be  noted  that  it  is  possible  for  two  variables  to  be 
closely  associated  or  correlated  and  yet  show  considerable  departure 
from  straight-line  relationship.  For  example,  if  strength  and  age  are 
compared  throughout  life,  it  is  found  that  when  persons  are  very  young 
their  strength  is  small,  that  as  they  become  older,  up  to  a  certain  limit, 
it  increases,  after  which  it  decreases  again.  That  is,  there  exists  a  fairly 
close  relationship  but  not  a  rectilinear  one.  To  completely  measure 
such  situations  as  this,  an  expression  for  curved-line  or  curvilinear  rela- 
tionship is  needed.  The  one  most  commonly  used  is  the  ratio  of  corre- 
lation,2 the  discussion  of  which  is  outside  the  scope  of  this  bulletin. 

One  may,  for  at  least  two  reasons,  think  of  the  coefficient  of  correla- 
tion as  a  minimum  measure  of  the  amount  of  relationship  existing.     In 


2See: 

Odell,  op.  cit.,  p.  207-13. 


[36] 


the  first  place,  if  the  association  is  rectilinear  it  measures  all  of  it,  but 
not  more,  whereas  if  the  association  is  at  all  curvilinear  it  measures 
somewhat  less  than  all  of  it.  Secondly,  in  practically  all  cases  variable 
or  chance  errors  enter  into  the  measurements  of  the  two  characteristics 
being  correlated  and  the  total  effect  of  these  errors,  called  attenuation,3 
is  to  make  the  computed  or  apparent  coefficient  of  correlation  less  than 
the  true  one.  If  two  series  of  similar  measurements  of  each  character- 
istic are  available,  it  is  possible  to  correct  for  the  effect  of  such  errors. 
When  these  are  not  available,  all  we  can  say  is  that  the  true  value  of  r 
is  as  great  as,  or  greater  than,  the  one  actually  obtained. 

As  was  stated  above  the  value  of  the  coefficient  of  correlation 
varies  from  +  1-00  through  zero  to  —  1.00.  A  value  of  +  1.00  indi- 
cates perfect  positive  correlation,  that  is,  that  each  score  in  one  distri- 
bution deviates  from  its  average  in  the  same  direction  and  by  the  same 
proportional  amount  as  does  the  corresponding  score  in  the  other  dis- 
tribution. A  coefficient  of  zero  indicates  that  there  is  no  correlation  or, 
in  other  words,  that  the  association  between  the  two  characteristics  is 
purely  a  chance  one.  If  r  equals  —  1.00,  the  two  variables  have  per- 
fect negative  correlation,  that  is,  each  score  in  one  series  deviates  from 
its  average  by  the  same  proportional  amount  as  the  corresponding  score 
of  the  other  series,  but  in  the  opposite  direction. 

The  coefficient  of  correlation  as  an  index  of  the  existence  or  ab- 
sence of  relationship.  Two  chief  purposes  for  determining  the  value 
of  the  coefficient  of  correlation  may  be  distinguished,  although  in  many 
cases  some  element  of  both  is  present.  One  of  these  purposes  is  to 
learn  whether  there  is  any  relationship  at  all  between  the  two  sets  of 
paired  facts  under  consideration.  In  other  words,  the  question  to  which 
one  is  seeking  an  answer  is  "Are  these  two  characteristics  related  to 
each  other?"  rather  than,  "How  close  is  the  relationship  between  these 
characteristics?"  One  may,  for  example,  desire  to  determine  whether 
or  not  any  relationship  exists  between  mental  and  physical  ability,  or 
between  ability  in  arithmetical  computation  and  in  solving  reasoning 
problems.  In  such  cases,  the  fact  that  the  coefficient  of  correlation  is 
found  to  be  appreciably  greater  than  zero  may  be  considered  as  evi- 
dence that  there  is  some  relationship  present. 

In  interpreting  a  coefficient  of  correlation  used  for  this  purpose, 
one  must  know  whether  the  obtained  value  of  r  is  being  considered  as  a 
measure  of  the  correlation  existing  between  the  particular  sets  of  paired 
facts  for  which  it  was  computed,  and  these  only,  or  whether  these  cases 


3See: 

Odell,  op.  cit.,  p-  181-85. 


[37] 


TABLE  IX.    CORRELATION  BETWEEN  SPEED  AND  QUALITY  OF 
WRITING  OF  CHILDREN  OF  SCHOOL  C 


Grade 

IV 

V 

VI 

VII 

VIII 

Coefficient 

of 
Correlation 

+.08(±.02) 

-.10(±.04) 

-14.(±.04) 

• 
-.34a 

-.15(±.05) 

aFreeman  does  not  give  the  P.E.  of  this  coefficient. 

are  to  be  considered  merely  as  a  sample  of  a  larger  number.  If  the 
latter  is  the  case  the  obtained  value  is  subject  to  errors  of  sampling  just 
as  is  any  other  derived  measure  similarly  used.  As  was  explained 
earlier4  in  this  bulletin,  the  most  usual  means  of  interpreting  an  obtained 
value  of  r  when  it  is  subject  to  errors  of  sampling  is  to  compare  it  with 
its  probable  error.  If,  on  the  other  hand,  one  is  concerned  merely  with 
the  cases  actually  measured  and  assumes  that  the  measurements  are 
accurate  and  the  computations  reliable,  there  is  no  need  for  interpreting 
r  by  comparison  with  such  a  measure  of  the  reliability  of  sampling. 

As  was  stated  above,  in  many  if  not  most  cases  in  which  the  value 
of  r  is  determined,  the  person  or  persons  doing  so  do  not  make  clear, 
and  perhaps  usually  do  not  have  definitely  in  mind,  which  one  of  the 
two  purposes  is  predominant.  In  other  cases,  however,  it  is  evident  that 
one  or  the  other  is  the  more  important  in  the  particular  situation.  An 
example  in  which  the  purpose  already  described  appears  to  be  predom- 
inant may  be  found  in  an  article  by  Freeman,5  in  which  he  gives  co- 
efficients of  correlation  between  speed  and  quality  of  handwriting  in 
Grades  IV  to  \  III.  Freeman's  purpose  was  undoubtedly  to  present 
evidence  as  to  whether,  in  general,  there  exists  any  relationship  between 
quality  and  speed  in  handwriting,  and  therefore  he  gave  the  probable 
errors  of  all  except  the  largest  coefficient.     It  will  be  seen  that  the  four 


4See  p.  21  for  a  discussion  of  the  probable  error  of  sampling. 

*Freeman,  F.  N.  "Some  practical  studies  of  handwriting,"  Elementary  School 
Teacher,  14:167-79,  December,  1913.  Other  examples  of  the  same  use  of  r  may  be 
found  in  the  following  references: 

Starch,  Daniel.  Educational  Psychology.  New  York:  The  Macmillan  Com- 
pany, 1923,  p.  246- 

Abernethy,  Ethel  M.  "Correlations  in  physical  and  mental  growth,"  Journal  of 
Educational  Psychology,  16:456-66,  October,  1925. 

Chapin,  F.  Stuart.  "Extra-curricular  activities  of  college  students:  a  study  in 
college  leadership,"  School  and  Society.  23:212-16,  February'  13,  1926. 

Furfey,  Paul  H.  "Some  preliminary  results  on  the  nature  of  developmental  age," 
School  and  Society,  23 :  183-84,  February  6,  1926. 


[38] 


small  coefficients  vary  from  two  and  one-half  to  four  times  their  prob- 
able errors.  Since  it  is  customary  to  consider  a  value  of  r  as  fairly  re- 
liable if  it  is  three  times  as  large  as  its  P.E.  and  to  consider  it  as  almost 
certainly  reliable  if  it  is  four  or  five  times  its  P.E.,  it  appears  that  the 
four  small  coefficients  are  probably  reliable,  but  that  only  the  one  for 
Grade  VII  can  certainly  be  said  to  be  so.  In  other  words,  the  data 
from  Grades  IV,  V,  VI  and  VIII  indicate  that,  when  all  pupils  are  con- 
sidered, speed  and  quality  of  handwriting  probably  are  slightly  nega- 
tively associated,  whereas  those  for  Grade  VII  seem  to  show  that  there 
is  no  doubt  of  the  existence  of  negative  relationship. 

To  illustrate  further  the  interpretation  of  r  when  used  for  this  pur- 
pose, let  us  suppose  that  the  correlation  between  the  average  number 
of  pupils  to  the  teacher  and  the  average  salaries  of  teachers  has  been 
found  for  all  cities  of  more  than  100,000  population  and  also  for  a  ran- 
dom sample  of  fifty  cities  having  from  50,000  to  100,000  population. 
Furthermore,  let  us  suppose  that  in  both  cases  the  obtained  value  of  r 
is  .20.  For  the  cities  above  100,000  such  a  value  would  indicate  defin- 
itely that  there  was  a  real,  though  small,  relationship  between  the  aver- 
age number  of  pupils  and  average  salaries.  This  is  true  because  all 
cities  of  the  class  were  included  and  there  was  no  sampling.  On  the 
other  hand,  for  the  cities  of  from  50,000  to  100,000  the  value  of  r  is  sub- 
ject to  a  P.E.  of  .096  because  in  this  case  a  sampling  was  made.  There- 
fore, it  is  probable  but  by  no  means  certain  that  some  correlation  really 
exists  for  all  cities  of  this  size,  the  chances  being  about  6  to  1  in  its 
favor.  If  a  sample  of  one  hundred  instead  of  fifty  cities  were  taken, 
the  P.E.  of  r  would  be  reduced  from  .09  to  .06  and  we  might  say  it  was 
fairly  certain  that  the  correlation  for  the  whole  group  was  positive,  since 
the  chances  are  about  40  to  1  in  its  favor.7  If  the  size  of  the  sample 
was  increased  still  more  the  probable  error  of  r  would,  of  course,  be 
still  further  reduced  and  the  chances  that  r  is  certainly  reliable  propor- 
tionately increased. 

The  coefficient  of  correlation  as  a  measure  of  the  closeness  of  re- 
lationship or  reliability  of  prediction.  The  second  purpose  for  which 
the  coefficient  of  correlation  is  commonly  used  is  to  indicate  just  how 
close  is  the  relationship  or  association  between  two  characteristics  or 
how  accurately  one  can  be  estimated  or  predicted  when  the  other  is 
known.     This  purpose  really  assumes  that  there  is  some  definite  rela- 


6The    formula    for    the    probable    error    of    the    coefficient    of    correlation    is 
1   -  r2  1   -  .202 

.6745 — 7=~.      Hence,  in  this  case  P.E.    =   .6745 — 7= —   =  .09. 

_VjV  r  V50 

'See  p.  14  of  this  bulletin. 

[39] 


tionship,  either  positive  or  negative,  and  seeks  to  determine  how  nearly 
it  approaches  complete  or  perfect  association  or,  in  other  words,  what 
the  probable  accuracy  of  estimating  one  variable  from  the  other  is.  The 
value  of  r  gives  a  measure  of  the  accuracy  or  reliability  of  the  prediction 
or  prognosis  possible.  For  example,  if  we  know  that  the  coefficient  of 
correlation  between  height  and  weight  is  .75,  we  have  some  idea  as  tc 
how  closely  a  given  person's  height  can  be  estimated  if  his  weight  is 
known,  or  vice  versa.  How  definite  an  idea  we  have  depends  largely 
upon  the  amount  of  our  experience  in  meeting  and  dealing  with  similar 
situations  involving  various  values  of  the  coefficient.  The  name  "co- 
efficient of  reliability"  or  "coefficient  of  self-correlation"  is  frequently 
applied  to  the  coefficient  of  correlation  between  two  series  of  duplicate 
measurements  of  the  same  individuals,  such  as  those  yielded  by  dupli- 
cate forms  of  a  test  or  measurements  of  height  by  two  persons.  Some- 
times these  names  are  also  applied  to  coefficients  of  correlation  between 
two  series  of  similar  but  not  duplicate  measurements,  such  as  those 
yielded  by  two  different  intelligence  tests  or  by  two  reading  tests.  For 
example,  if  pupils'  abilities  in  the  fundamental  operations  of  arithmetic 
are  measured  by  Form  1  of  the  Courtis  Research  Tests  in  Arithmetic, 
Series  B,  and  later  this  is  repeated,  or  one  of  the  equivalent  forms  used, 
the  coefficient  of  correlation  between  the  two  series  of  scores  is  the  co- 
efficient of  reliability.  Likewise,  the  term  is  often  though  less  frequently 
applied  to  the  correlation  between  the  scores  of  a  group  of  pupils  on, 
for  example,  the  National  Intelligence  Tests  and  the  Illinois  General 
Intelligence  Scale. 

An  example  of  the  use  of  the  coefficient  of  correlation  with  this 
purpose  predominating  is  shown  by  Table  X,  prepared  by  Starch.8 
It  contains  the  coefficients  of  correlation  found  between  school  marks  of 
two  groups  of  pupils  in  various  subjects.  This  table  shows,  for  example, 
that  a  pupil's  grade  in  arithmetic  can  be  more  closely  predicted  from 


8Starch,  Daniel.  "Correlation  among  abilities  in  school  studies,  Journal  of  Edu- 
cational Phychology,  3:415-18,  September,  1913.  Other  examples  may  be  found  by 
consulting  the  following: 

Orleans,  J.  S.  "The  ability  to  spell/'  School  and  Society,  23:407-08,  March  27, 
1926. 

Nanninga,  S.  P.  "A  critical  study  of  rating  traits,"  Educational  Administration 
and  Supervision,  12:114-19,  February,  1926. 

Hull,  C.  L.,  and  Limp,  C.  E.  "The  differentiation  of  the  aptitudes  of  an  individ- 
ual by  means  of  test  batteries,"  Journal  of  Educational  Psychology,  16:73-88,  February. 
1925. 

Ruch,  G.  M.,  and  Stoddard,  G.  D.  '"Comparative  reliabilities  of  five  types  of  ob- 
jective examinations,"  Journal  of  Educational  Psychology,  16:89-103,  February,  1925. 

[40] 


TABLE  X.     COEFFICIENTS  OF   CORRELATION  BETWEEN   MARKS  OF 
TWO  GROUPS  OF  PUPILS  IN  SEVERAL  SCHOOL  SUBJECTS 


Second 
Group 


Arithmetic  and  language. 
Arithmetic  and  geography 
Arithmetic  and  history. . . 
Arithmetic  and  reading. . . 
Arithmetic  and  spelling. . . 
Language  and  geography. 
Language  and  history. . . . 
Language  and  reading. . . . 
Language  and  spelling. . . . 
Geography  and  history. . . 
Geography  and  reading. . . 
Geography  and  spelling.  . 

History  and  reading 

History  and  spelling 

Reading  and  spelling 


.  /J 
.74 

.73 
.45 
.42 


.77 
.81 
.83 

68 
,67 

37 
.72 


58 


his  grade  in  language,  with  which  the  correlations  are  .73  and  .85,  than 
from  his  grade  in  spelling,  which  correlates  with  it  only  .42  and  .55. 

In  connection  with  the  use  of  r  for  this  purpose  one  should  bear  in 
mind  that  its  value  may  be  large  enough  to  indicate  that  there  is  a 
definite  association  between  the  two  characteristics  correlated  and  yet 
not  large  enough  to  enable  one  to  place  much  confidence  in  the  predic- 
tion of  the  probable  amount  of  one  trait  possessed  by  an  individual 
when  that  of  the  other  is  known.  Furthermore^  the  value  of  r  in  itself 
does  not  give  a  direct  measure  of  the  size  of  the  errors  liable  to  be  pres- 
ent in  predictions  or  estimates  based  upon  the  data  from  which  r  was 
computed.  It  is,  therefore,  frequently  desirable  to  interpret  coefficients 
of  reliability  and  other  coefficients  of  correlation  used  for  estimating  one 
characteristic  from  another  by  finding  the  probable  errors  of  estimate 
and  of  measurement^  associated  with  them.  The  following  paragraph 
will  describe  the  method  of  doing  so. 

Interpretation  of  the  coefficient  of  correlation  in  terms  of  the 
probable  errors  of  estimate  and  of  measurement.  The  formulae  for  the 
probable  errors  of  estimate  and  of  measurement  which  were  given  on 
p.  29  show  that  their  magnitude  depends  upon  two  things — the  coeffi- 
cient of  correlation  and  the  median  deviation  of  the  distribution.  We 
can  therefore  easily  find  for  any  given  value  of  r  the  size  of  the  prob- 
able errors  of  estimate  and  of  measurement  in  terms  of  Md.D.  as  the 


8"For  a  discussion  of  these  measures  see  above,  p.  28  et  seq. 

[41] 


TABLE  XI.   VALUES  OF  THE  PROBABLE  ERRORS  OF  ESTIMATE  AND  OF 
MEASUREMENT    CORRESPONDING    TO    CERTAIN    VALUES 
OF   THE    COEFFICIENT   OF    CORRELATION 


Coefficient 

of 
Correlation 

Probable 

Error 

of 

Estimate 

Probable 

Error 

of 

Measurement 

1.00 

.0000 

Md. 

D. 

.0000 

Md. 

D. 

.99 

.1411 

Md. 

D. 

.1000 

Md. 

D. 

.98 

.1990 

Md. 

D. 

.1414 

Md. 

D. 

.97 

.2431 

Md. 

D. 

.1732 

Md. 

D. 

.96 

.2800 

Md. 

D. 

.2000 

Md. 

D. 

.95 

.3122 

Md. 

D. 

.2236 

Md. 

D. 

.90 

.4359 

Md. 

D. 

.3162 

Md. 

D. 

.80 

.6000 

Md. 

D. 

.44": 

Md. 

D. 

.70 

."141 

Md. 

D. 

.5477 

Md. 

D. 

.60 

.8000 

Md. 

D. 

.6325 

Md. 

D. 

.50 

.8660 

Md. 

D. 

.7071 

Md. 

D. 

.40 

.9165 

Md. 

D. 

.7746 

Md. 

D. 

.30 

.9539 

Md. 

D. 

.8367 

Md. 

D. 

.20 

.9798 

Md. 

D. 

.8944 

Md. 

D. 

.10 

.9950 

Md. 

D. 

.9487 

Md. 

D. 

.00 

1.0000 

Md. 

D. 

1.0000 

Md. 

D. 

unit.  Table  XI  has  been  inserted  to  give  the  probable  errors  of  estimate 
and  of  measurement  for  the  values  of  r  from  .00  to  .90  at  intervals  of 
.10,  and  from  .95  to  1.00  at  intervals  of  .01.  For  example,  if  the  coeffi- 
cient of  correlation  is  .99  the  probable  error  of  estimate  is  .1411  Md.D. 
and  that  of  measurement  .1000  Md.D.  Similarly,  if  r  =  .70,  P.E. 
=  .7141  Md.D.  and  P.E.  „      =  .5477  Md.D.    Glancing  over  the  whole 

Meas. 

table  one  sees  that  an  increase  of  the  same  amount  in  the  coefficient  of 
correlation  produces  a  greater  decrease  in  the  errors  when  r  is  high 
than  when  it  is  low. 

The  preceding  discussion  has  probably  not  made  absolutely  clear 
the  significance  of  a  probable  error  of  estimate  or  of  measurement  ex- 
pressed in  terms  of  Md.D.  The  following  statement  may  be  helpful  in 
this  connection.  When  r  =  .00,  or  in  other  words  when  no  correlation 
at  all  exists,  both  P.E.  r   and  P.E.  ,,       =  1.0000  Md.D.  This  means  that 

'  En.  Mens. 

if  one  attempted  to  estimate  scores  in  one  distribution  from  those  in  the 
other  by  making  pure  guesses."  he  might  expect  that  in  half  of  his  esti- 


?In  using  the  term  "pure  guesses"  it  is  understood  that  the  person  so  guessing 
knows  the  limits  and  the  general  shape  of  the  distribution  of  scores  being  guessed.  He 
does  not.  however,  have  at  his  command  any  information  whatsoever  which  helps  him 
in  guessing  the  location  of  any  particular  score  within  this  distribution. 

[42] 


mates  or  guesses  he  would  be  in  error  by  amounts  less  than  Md.D.,  and 
in  half  by  amounts  greater.  From  this  point  of  view  it  can  be  seen 
that  even  though  the  coefficient  of  correlation  is  rather  large  a  great 
deal  of  the  guessing  element  is  present  in  estimating  scores  in  one  dis- 
tribution from  those  in  the  other.  Many  people  commonly  think  that 
if  r  =  .85  or  .90  the  association  is  very  close  or  almost  perfect,  whereas 
as  a  matter  of  fact  an  estimate  is  still  half  a  pure  guess  when  r  =  .866, 
and  even  when  r  =  .968  an  estimate  is  one-fourth  a  pure  guess.  When 
one  is  estimating  a  true  score  from  a  score  actually  obtained  the  esti- 
mate is  half  a  guess  when  r=.75,  and  one-fourth  a  guess  when  r  is 
almost  .94.  Thus  it  can  be  seen  that  the  coefficient  of  correlation  must 
either  approach  1.00  very  closely,  or  equal  it,  before  the  errors  of  esti- 
mate and  of  measurement  are  small  enough  to  be  negligible.  On  the 
other  hand,  even  when  these  errors  are  considerable  they  are  less  than 
would  be  the  case  if  no  correlation  existed,  and  therefore  one  can  make 
better  estimates  of  scores  in  one  distribution  from  those  in  another  if 
there  is  any  correlation  at  all  between  the  two  than  he  can  if  no  helpful 
information  of  any  kind  is  available.10 

To  illustrate  still  more  clearly  the  meaning  of  the  probable  errors 
of  estimate  and  of  measurement  Figure  511  is  given.  It  shows  the  cor- 
relation between  scores  on  Forms  1  and  2  of  the  Illinois  General  In- 
telligence Scale.  This  figure  is  in  general  similar  to  Tables  VII  and 
VIII  except  that  instead  of  numbers  to  show  how  many  scores  fall  in 
each  cell  it  contains  a  dot  for  each  score.  The  height  of  each  dot  above 
the  base  line  (X-axis)  shows  the  Form  1  score  made  by  the  individual 
represented  by  the  dot,  and  its  distance  to  the  right  of  the  vertical  line 
at  the  left  of  the  table  (F-axis)  shows  the  Form  2  score  of  the  same  in- 
dividual. In  a  few  cases  figures  showing  these  distances  have  been 
placed  in  parenthesis  after  the  dots.  In  such  cases  the  first  of  these 
two  numbers  indicates  the  Form  2  score  or  X  distance  and  the  latter  of 
the  two  the  Form  1  score  or  Y  distance.  For  example,  near  the  upper 
right  hand  corner  of  the  figure  is  a  dot  representing  a  pupil  who  made 
a  score  of  104  on  Form  2  and  118  on  Form  1. 


10The  actual  procedure  of  estimating  scores  in  one  series  from  those  in  another 
with  which  it  is  correlated  involves  the  use  of  the  regression  equation,  which  is  based 
upon  the  averages  and  standard  deviations  of  the  two  series  and  the  coefficient  of  corre- 
lation.    For  an  explanation  of  regression,  see: 

Odell,  op.  cit.,  p.  189-96. 

"This  figure  is  taken  from: 

Monroe,  W.  S.  "The  Illinois  examination."  University  of  Illinois  Bulletin,  Vol. 
19.  Xo.  9,  Bureau  of  Educational  Research  Bulletin  No.  6.  Urbana:  University  of  Illi- 
nois, 1921.    45  p. 

[43] 


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£4- 


Figure  5.     Correlation  of  Form  1  Scores  with  Form  2  Scores  or 
the  Illinois  General  Intelligence  Scale.  Fifth  Grade 


The  long  diagonal  line  in  the  figure  is  the  Y  regression  line.  In 
other  words  it  is  a  graphical  representation  of  the  best  possible  straight- 
line  equation  for  estimating  the  Y  or  Form  1  score  when  the  X  or  Form 
_  re  is  known,  which  can  be  derived  from  the  correlation  between 
the  two  distributions  and  their  spread  or  variability.  If  the  correlation 
were  perfect  there  would  be  no  errors  in  such  estimates  and  all  the  dots 


[44] 


would  lie  upon  the  diagonal  line.  As  it  is  the  vertical  distance  from  each 
dot  to  this  line  represents  the  error  involved  in  estimating  the  Form  1 
score  from  the  Form  2  score.  In  several  cases  the  vertical  lines  con- 
necting the  dots  with  the  diagonal  have  been  drawn  in.  with  a  number 
beside  each  line  to  indicate  its  length,  or  in  other  words  the  error  of 
estimate  in  that  particular  case.  For  example,  the  error  of  estimate  for 
the  case  previously  mentioned  as  having  scores  of  104  and  118  is  50. 
Substituting  in  the  formula  given  above.12  it  is  found  that  P.E.         is 

e  e  •  Est. 

6.06.  Therefore  we  know  that  the  errors  involved  in  estimating  Form 
1  from  Form  2  scores  are  less  than  this  in  half  of  the  cases  and  greater 
in  the  other  half.  In  other  words,  the  vertical  distances  from  half  of  the 
dots  to  the  diagonal  line  are  less  than  6.06.  whereas  those  from  the  other 
half  are  greater  than  this  amount.  A  similar  diagonal  line  representing 
the  estimates  of  Form  2  scores  from  Form  1  scores  could  also  be  drawn. 
If  it  were,  the  horizontal  distances  from  the  points  to  that  line  would 
represent  the  errors  in  those  estimates. 

Interpretation  of  the  coefficient  of  correlation  by  comparison  with 
the  sizes  of  coefficients  of  correlation  commonly  found.  A  second 
means  of  interpreting  r  is  by  comparing  its  value  with  the  coefficients 
of  correlation  found  to  exist  in  certain  relatively  familiar  situations.  One 
has  a  more  or  less  definite  idea  of  the  extent  to  which  tall  people  tend 
to  weigh  more  and  short  people  less,  of  how  much  children  tend  to  re- 
semble their  parents  in  height,  and  so  on.  Therefore,  by  comparing  the 
value  of  a  coefficient  of  correlation  with  those  usually  found  in  some  of 
these  fairly  common  and  well-known  cases,  such  an  idea  may  be  formed 
as,  for  example,  that  the  relationship  in  question  is  somewhat  closer 
than  that  between  height  and  weight,  or  about  the  same  as  that  between 
school  marks  in  Latin  and  in  French.  It  is  true  that  our  ideas  as  to  just 
how  closely  two  characteristics,  even  though  they  are  very  common,  are 
related,  are  decidedly  subjective  and  therefore  often  considerably  in 
error,  yet  such  comparisons  have  some  value  in  interpreting  coefficients 
of  correlation.  To  aid  the  reader  in  making  such  interpretations  the 
following  table  showing  the  magnitude  of  the  coefficients  of  correlation 
usually  found  between  pairs  of  certain  fairly  common  characteristics 
is  given. 

Interpretation  of  r  in  terms  of  displacement.  Another  means  of 
interpreting  the  value  of  r  is  in  terms  of  the  differences  in  individuals' 
relative  positions  in  the  two  series  of  measures  correlated  or.  as  this  is 
commonly  called,  in  terms  of  displacement.    For  example,  if  height  and 

"See  p.  29. 

[45] 


TABLE  XII.     SIZES  OF   COEFFICIENTS  OF   CORRELATION* 
COMMONLY  FOUND 


Cost  of  instruction  with  total  cost  of  education 

First  and  second  applications  of  an  individual  intelligence  test 

Ages  of  husbands  with  ages  of  wives 

First  and  second  applications  of  a  standardized  group  test 

School  marks  in  subjects  supposed  to  be  more  or  less  akin,  such  as  Eng- 
lish and  foreign  language,  or  mathematics  and  physics 

Heights  of  fathers  with  heights  of  sons 

School  marks  in  subjects  supposed  to  have  little  in  common,  such  as 
Latin  and  domestic  science,  or  English  and  manual  training 

Quality  of  handwriting  and  intelligence  test  scores 


.90-. 95 
.90-. 95 
.85-. 95 
.60-. 90 

.40-. 70 
.40-. 60 

.30-. 50 
.00-. 10 


weight  are  the  two  characteristics  concerned  and  a  certain  individual  is 
fourth  from  the  top  in  height,  his  displacement  is  the  amount  that  he 
differs  from  this  position  in  weight.  The  interpretation  is  usually  made 
by  expressing  the  probability  that  position  or  rank  in  one  variable  does 
not  differ  by  more  than  a  certain  distance  or  number  of  places  from 
that  in  the  other.  A  table  which  may  be  used  for  this  purpose  has  been 
prepared  and  published  by  Otis,13  but  is  not  reproduced  here  because 
the  interpretation  of  r  through  the  amount  of  displacement  has  not  come 
into  common  use.  Also  it  is  somewhat  difficult  to  comprehend  readily 
just  what  is  meant  by  this  method  of  interpretation. 

The  interpretation  of  the  coefficient  of  correlation  in  terms  of 
adjectives.  Some  writers  have  undertaken  to  define  the  meaning  of  coeffi- 
cients of  correlation  by  means  of  certain  adjectives  which  they  apply  to 
coefficients  of  various  sizes.  Rugg,14  for  example,  states  that  his  experi- 
ence "has  led  him  to  regard  correlation  as  "negligible'  or  "indifferent' 
when  r  is  less  than  .15  to  .20;  as  being  'present  but  low'  when  r  ranges 
from  .15  or  .20  to  .35  or  .40;  as  being  'markedly  present'  or  'marked' 
when  ;•  ranges  from  .35  or  .40  to  .50  or  .60;  as  being  'high'  when  it  is 
above  .60  or  .70.  With  the  present  limitations  on  educational  testing 
few  correlations  in  testing  will  run  above  .70,  and  it  is  safe  to  regard  this 
as  a  very  high  coefficient."  McCall15  likewise  offers  a  statement  of  this 
sort,  but  briefer  than  Rugg's,  as  follows : 


13Oris,  A.  S.  Statistical  Method  in  Educational  Measurement.  Yonkers:  World 
Book  Company.  1925.  p.  225. 

"Rugg,  H.  O.  Statistical  Methods  Applied  to  Education.  Boston:  Houghton 
Mifflin  Company,  1917,  p.  256.    Also  see: 

Rugg,  H.  0.  A  Primer  of  Graphics  and  Statistics  for  Teachers.  Boston:  Hough- 
ton Mifflin  Company,  1925,  p.  97. 

'"McCall,  \V.  A.  How  to  Measure  in  Education.  New  York:  The  Macmillan 
Company,   1922,   p.  392-93. 

[46] 


''when  r  is  0  to  ±  A  correlation  is  low,  or 

±  A  to  ±     .7  correlation  is  substantial,  or 
±  .7  to  ±  1.0  correlation  is  high." 

The  chief  purpose  of  the  present  writer  in  mentioning  this  method 
of  interpreting  r  is  to  point  out  that  the  use  of  adjectives  is  decidedly 
unsatisfactory  and  indeed  often  meaningless  unless  they  are  employed 
in  a  definitely  limited  situation.  Whether  a  coefficient  of  correlation  is 
high  or  fair  or  low  depends  upon  the  purpose  for  which  it  is  employed 
and  the  data  for  which  it  is  computed.  A  coefficient  of  .30  or  .40,  for 
example,  is  high  enough  to  indicate  that  there  is  definite  relationship  be- 
tween the  two  things  correlated  but  at  the  same  time  it  is  so  low  that, 
as  has  been  shown,  estimates  of  one  of  the  traits  from  the  other  are 
scarcely  better  than  mere  guesses.  Again,  a  correlation  of  .80  between 
school  marks  in  chemistry  and  in  physics  is  relatively  high  since  the 
usual  correlation  between  such  marks  is  considerably  lower  than  this, 
but  a  correlation  of  the  same  size  between  two  applications  of  the  same 
individual  intelligence  test  is  low  since  the  best  of  such  tests  yield  cor- 
relations of  .90,  or  above.  The  writer  therefore  wishes  to  repeat  that 
it  is  very  undesirable  to  describe  the  amount  of  correlation  by  means  of 
adjectives  unless  it  is  done  in  view  of  a  definite  and  particular  situation. 

Effect  of  spread  of  data  upon  value  of  r.  Another  topic  that  should 
be  treated  is  the  interpretation  of  the  coefficient  of  correlation  in  view 
of  certain  facts  concerning  the  data  for  which  it  is  computed.  One  of 
these  important  facts  which  should  be  known  is  the  spread  of  data,  that 
is,  the  extent  to  which  they  vary  or  scatter  away  from  their  average. 
Probably  the  most  common  occasion  on  which  this  is  important  is  when 
there  is  a  difference  in  the  number  of  school  grades  that  contributed  the 
data  for  the  two  or  more  correlations  being  compared.  Frequently,  co- 
efficients of  correlation  are  determined  between  series  of  measurements 
obtained  from  a  single  grade,  whereas  on  other  occasions  they  are  based 
upon  those  from  several  grades.  In  many  cases  the  spread  of  the  char- 
acteristic measured  increases  as  the  number  of  grades  is  increased.  For 
example,  the  variations  in  height,  weight,  mental  age,  score  upon  a 
subject-matter  test,  and  so  forth,  may  be  expected  to  be  greater  in  twc 
grades  than  in  one,  greater  in  three  than  in  two,  and  so  on.  The  effect 
of  this  increased  spread  is  to  raise  the  obtained  value  of  the  coefficient 
of  correlation  although,  of  course,  the  degree  of  relationship  is  not 
changed.  For  example,  the  correlation  between  age  and  height  may 
average  only  .40  in  each  grade  but  if  all  grades  from  one  to  eight  are 
included,  it  will  probably  be  at  least  .70  or  .80.  Sometimes  the  effect  of 
increasing  the  spread  is  so  pronounced  that  correlations  which  are  nega- 

[47] 


tive  for  a  single  grade  or  other  limited  group  become  positive  for  a 
more  scattered  or  variable  group.  One  of  the  common  examples  of  this 
is  the  correlation  between  chronological  and  mental  age.  Within  any 
given  grade  it  is  almost  always  true  that  the  younger  pupils  are  the 
brighter  and  the  older  ones  the  duller,  so  that  the  correlation  between 
mental  and  chronological  age  in  a  single  grade  is  almost  always  nega- 
tive, generally  from  —  .20  to  —  .50.  If  two  or  three  grades  are  taken 
together  this  correlation  usually  changes  to  about  zero,  whereas  if  five 
or  six  are  included  it  becomes  positive,  probably  from  .50  to  .70,  agree- 
ing with  the  fact  that  older  children  tend  to  have  higher  mental  ages 
than  younger  ones. 

Because  of  this  effect  of  the  spread  of  the  group  upon  the  value  of 
r,  any  given  value  thereof  should  be  accompanied  by  a  statement  defin- 
ing the  group  for  which  it  was  computed.  It  is  also  frequently  desirable 
to  give  some  measure,  such  as  the  median  or  standard  deviation,  of  the 
spread  of  each  group.  By  means  of  a  formula16  not  given  here,  one  can 
then  make  allowance  for  the  effect  of  different  degrees  of  spread  upon 
the  coefficient  of  correlation,  and  thus  compare  different  values  of  r  upon 
a  true  basis. 

It  should  be  noted  that  one  group  may  have  a  greater  spread  than 
another  in  some  characteristic  or  characteristics  other  than  those  cor- 
related without  affecting  the  value  of  r.  For  example,  the  range  or 
spread  of  intelligence  quotients  in  an  average  group  of  pupils  from  sev- 
eral grades  is  little,  if  any,  greater  than  in  a  group  from  one  grade  only, 
altho  the  spread  of  the  pupils  as  regards  grade,  age,  and  so  on,  is  much 
greater  in  the  first  group.  The  same  is  true  of  the  school  marks  given 
by  teachers,  of  health  ratings  of  teachers'  salaries,  and  so  on.  There- 
fore, in  cases  such  as  these  it  is  not  necessary  to  allow  for  the  fact  that 
several  grade  groups  instead  of  one  are  included.  For  example,  the 
correlation  between  /.O's  and  school  marks  for  a  group  from  several 
grades  would  be  practically  the  same  as  for  a  single  grade  group.  On 
the  other  hand,  it  may  be  that  there  is  a  difference  in  the  spread  of  these 
characteristics  due  to  some  less  common  basis  of  grouping  than  grades. 
If,  for  example,  pupils  have  been  grouped  according  to  their  mental 
ability  the  spread  of  I.O.'s  will  be  greater  in  a  combined  group  embrac- 
ing sections  of  various  abilities  than  in  a  single  group  of  bright,  average, 
or  dull  pupils. 

Averaging  coefficients  of  correlation.  One  not  infrequently  sees 
such  a  statement  as  that  the  average  coefficient  of  correlation  is  .60  or 


16This  is  given  in: 

Odell,  op.  cit.,  p.  174-77. 


[48] 


.85,  for  example.  The  process  of  averaging  coefficients  of  correlation  is 
not  one  which  should  be  indulged  in  without  taking  precautions  that 
the  average  so  obtained  is  statistically  justified.  To  illustrate  this,  if 
the  correlation  between  intelligence  and  reading  ability  is  .40  for  one 
group  of  pupils  and  .60  for  another,  it  is  only  by  chance  that  it  will  be 
.50  if  the  data  for  the  two  groups  are  thrown  into  one  correlation  table. 
To  insure  this  result  the  numbers  of  cases  in  the  two  groups,  the  aver- 
ages and  the  spreads  of  the  two  groups  around  their  averages,  must  be 
the  same.  These  conditions  are  very  rarely  fulfilled.  For  practical 
purposes,  however,  if  the  averages  and  spreads  are  not  very  different 
and  if  each  correlation  is  weighted  by  the  number  of  cases  which  con- 
tribute to  it,  the  average  obtained  may  be  considered  as  fairly  repre- 
sentative. It  is,  however,  usually  if  not  always  much  better  to  give  all 
the  obtained  values  of  r  than  to  give  merely  their  average.  Certainly, 
if  the  average  is  given  it  should  be  made  clear  that  it  is  only  a  more  or 
less  rough  or  approximate  estimate  of  the  amount  of  correlation. 


THE  LIBRARY  OF  fHr 

MAR  I  4  m7 

'JOTJRSSTY  0*  ILLINOIS 


[49] 


'/' 


____»_ 


